Ad Code

Responsive Advertisement

Vector Algebra Class 12 Maths Important Questions Chapter 10

 

Vector Algebra Class 12 Important Questions with Solutions Previous Year Questions

Algebra of Vectors

Question 1.
Find the position vector of a point which divides the join of points with position vectors a⃗ 2b⃗  and 22a⃗ +b⃗  externally in the ratio 2:1. (Delhi 2016)
Answer:
Let given position vectors are OA=a⃗ 2b⃗  and OB=2a⃗ +b⃗ .

Let OA be the position vector of a point C which divides the join of points, with position vectors
OA and OB, externally in the ratio 2:1.
∴ OA = 2OB1OA21=2(2a⃗ +b⃗ )1(a⃗ 2b⃗ )1 [by external section formula]
= 4a⃗  + 2b⃗  – a⃗  + 2b⃗  = 3a⃗  + 4b⃗ 

Question 2.
If a⃗  = 4î – ĵ + k̂ and b⃗  = 2î – 2ĵ + k̂, then find a unit vector parallel to the vector a⃗ +b⃗ . (All India 2016)
Answer:
Given vectors are
a⃗  = 4î – ĵ + k̂, b⃗  = 2î – 2ĵ + k̂.
Now, a⃗ +b⃗  == (4î – ĵ + k̂) + (2î – 2ĵ + k̂)
= 6î – 3ĵ + 2k̂

and |a⃗ +b⃗ |=(6)2+(3)2+(2)2
36+9+4=49 = 7units

∴ The unit vector parallelto the vector a⃗ +b⃗  is
a⃗ +b⃗ |a⃗ +b⃗ |=6i^3j^+2k^7

Question 3.
The two vectors ĵ + k̂ and 3î – ĵ + 4k̂ represent the two sides AB and AC respectively of triangle ABC. Find the length of the median through A. (Delhi 2016; Foreign 2015)
Answer:
Given, AB = ĵ + k̂ and AC = 3î – ĵ + 4k̂
Vector Algebra Class 12 Maths Important Questions Chapter 10 1
Vector Algebra Class 12 Maths Important Questions Chapter 10 2

Alternate Method:
Given AB = ĵ + k̂ and AC = 3î – ĵ + 4k̂
Vector Algebra Class 12 Maths Important Questions Chapter 10 3

Question 4.
Write the direction ratios of the vector 3a⃗  + 2b⃗ , where a⃗  = î + ĵ – 2k̂ and b⃗  = 2î – 4ĵ + 5k̂ (All India 2015C)
Answer:
Clearly, 3a⃗  + 2b⃗  = 3 (î + ĵ – 2k̂) + 2 (2î – 4ĵ + 5k̂)
= (3î + 3ĵ – 6k̂) + (4î – 8ĵ + 10k̂)
= 7î – 5ĵ + 4k̂
Hence, direction ratios of vectors 3a⃗  + 2b⃗  are 7, – 5 and 4.

Question 5.
Find the unit vector in the direction of the sum of the vectors 2î + 3ĵ – k̂ and 4î – 3ĵ + 2k̂. (Foreign 2015)
Answer:
Let a⃗  = 2î + 3ĵ – k̂ and b⃗  = 4î – 3ĵ + 2k̂
Now, sum of two vectors,
a⃗ +b⃗  = (2î + 3ĵ – k̂) + (4î – 3ĵ + 2k̂) = 6î + k̂
Vector Algebra Class 12 Maths Important Questions Chapter 10 4

Question 6.
Find a vector in the direction of vector 2î – 3ĵ + 6k̂ which has magnitude 21 units. (Foreign 2014)
Answer:
To find a vector in the direction of given vector, first of all we find unit vector in the direction of given vector and then multiply it with given magnitude.

Let a⃗  = 2î – 3ĵ + 6k̂
Then, |a⃗ | = (2)2+(3)2+(6)2
4+9+36=49 = 7 units

The unit vector in the direction of the given vector a⃗  is
Vector Algebra Class 12 Maths Important Questions Chapter 10 5

Now, the vector of magnitude equal to 21 units
and in the direction of a is given by
21â = 21(27i^37j^+67k^) = 6î – 9ĵ + 18k̂

Question 7.
Find a vector a of magnitude 5√2, making an angle of Ï€4 with X-axis, Ï€2 with Y-axis and an acute angle 0 with Z-axis. (All India 2014)
Answer:
Here, we have l = cos Ï€4, m = cos Ï€2 and n = cosθ
⇒ l = 12, m = 0 and n = cosθ
Vector Algebra Class 12 Maths Important Questions Chapter 10 6

Question 8.
Write a unit vector in the direction of the sum of the vectors a⃗  = 2î + 2ĵ – 5k̂ and b⃗  = -2î + ĵ – 7k̂. (Delhi2014C)
Answer:
113(4î + 3ĵ – 12k̂)

Question 9.
Find the value of p for which the vectors 3î + 2ĵ + 9k̂ and î – 2pĵ + 3k̂ are parallel. (All India 2014)
Answer:
Given, 3î + 2ĵ + 9k̂ and î – 2pĵ + 3k̂ are two parallel vectors, so their direction ratios will be proportional.
Vector Algebra Class 12 Maths Important Questions Chapter 10 7

Question 10.
Write the value of cosine of the angle which the vector a⃗  = î + ĵ + k̂ makes with Y-axis. (Delhi 2014C)
Answer:
Given, a⃗  = î + ĵ + k̂
Now, unit vector in the direction of a⃗  is
Vector Algebra Class 12 Maths Important Questions Chapter 10 8
∴ Cosine of the angle which given vector makes with Z-axis is 13

Question 11.
Find the angle between X-axis and the vector î + ĵ + k̂. (All India 2014C)
Answer:
Let a⃗  = î + ĵ + k̂
Now, unit vector in the direction of a⃗  is
Vector Algebra Class 12 Maths Important Questions Chapter 10 9
So, angle between X-axis and the vector
î + ĵ + k̂ is cos α = 13 ⇒ α = cos-1 (13)
[∵ â = lî + mĵ + nk̂ and cos α = l ⇒ α = cos-1l]

Question 12.
Write a vector in the direction of the vector î – 2ĵ + 2k̂ that has magnitude 9 units. (Delhi 2014C)
Answer:
3î – 6ĵ + 6k̂

Question 13.
Write a unit vector in the direction of vector PQ, where P and Q are the points (1, 3, 0) and (4, 5, 6), respectively. (Foreign 2014)
Answer:
First, find the vector PQ by using the formula (x2 – x1)î + (y2 – y1)ĵ + (z2 – z1)k̂, then required unit vector is given by PQ|PQ|

Given points are P (1, 3, 0) and Q (4, 5, 6).
Here, x1 = 1, y1 = 3, z1 = 0
and x2 = 4, y2 = 5, z2 = 6
So, vector PQ = (x2 – x1)k̂ + (y2 – y1)ĵ + (z2 – z1)k̂
= (4 – 1)î + (5 – 3)ĵ + (6 – 0)k̂
= 3î + 2ĵ + 6k̂
∴ Magnitude of given vector
Vector Algebra Class 12 Maths Important Questions Chapter 10 10
Hence, the unit vector in the direction of PQ is
Vector Algebra Class 12 Maths Important Questions Chapter 10 11

Question 14.
If a unit vector a⃗  makes angle Ï€3 with î, Ï€4 with ĵ and an acute angle θ with k̂, then find the value of θ. (Delhi 2013)
Answer:
Here, we have
l = cosÏ€3, m = cos Ï€4 and n = cos θ
Vector Algebra Class 12 Maths Important Questions Chapter 10 12

Question 15.
Write a unit vector in the direction of the sum of vectors a⃗  = 2î – ĵ + 2k̂ and b⃗  = – î + ĵ + 3k̂. (Delhi 2013)
Answer:
126i^+526k^

Question 16.
If a⃗  = xî +2ĵ – zk̂ and b⃗  = 3î – yĵ + k̂ are two equal vectors, then write the value of x + y + z. (Delhi 2013)
Answer:
Two vectors are equal, if coefficients of their components are equal.
Given, a⃗ =b⃗  ⇒ xî + 2ĵ – zk̂ = î – yĵ + k̂

On comparing the coefficient of components, we get
x = 3, y = -2, z = -1
Now, x + y + z = 3 – 2 – 1 = 0

Question 17.
P and Q are two points with position vectors 3a⃗  – 2b⃗  and a⃗  + b⃗ , respectively. Write the position vector of a point R which divides the line segment PQ in the ratio 2 : 1 externally. (All India 2013)
Answer:
a⃗ +4b⃗ 

Question 18.
L and M are two points with position vectors 2a⃗  – b⃗  and a⃗  + 2b⃗ , respectively. Write the position vector of a point N which divides the line segment LM in the ratio 2 : 1 externally. (All India 2013)
Answer:
5b⃗ 

Question 19.
A and B are two points with position vectors 2a⃗  – 3b⃗  and 6b⃗  – a⃗ , respectively. Write the position vector of a point P which divides the line segment AB internally in the ratio 1:2. (All India 2013)
Answer:
Given, A and B are two points with position vectors 2a⃗  – 3b⃗  and 6b⃗  – a⃗ , respectively. Also, point P divides the line segment AB in the ratio 1 : 2 internally.
Vector Algebra Class 12 Maths Important Questions Chapter 10 13

Question 20.
Find the sum of the vectors a⃗  = î – 2ĵ + k̂ b⃗  = – 2î + 4ĵ + 5k̂ and c⃗  = î – 6ĵ – 7k̂. (Delhi 2012)
Answer:
Given vectors are a⃗  = î – 2ĵ + k̂ b⃗  = – 2î + 4ĵ + 5k̂ and c⃗  = î – 6ĵ – 7k̂.
Sum of the vectors a⃗ b⃗  and c⃗  is
a⃗ +b⃗ +c⃗  = (î – 2ĵ + k̂) + (- 2î + 4ĵ + 5k̂) + (î – 6ĵ – 7 k̂)
= – 4ĵ – k̂

Question 21.
Find the sum of the following vectors. a⃗  = î – 3k̂, b⃗  = 2ĵ – k̂, c⃗  = 2î – 3ĵ + 2k̂. (Delhi 2012)
Answer:
3î – ĵ – 2k̂

Question 22.
Find the sum of the following vectors. a⃗  = î – 2ĵ, b⃗  = 2î – 3 ĵ, c⃗  = 2î + 3k̂. (Deihi 2012)
Answer:
5î – 5ĵ + 3k̂

Question 23.
Find the scalar components of AB with initial point A (2,1) and terminal point B(- 5, 7). (All India 2012)
Answer:
Given initial point is A (2,1) and terminal point is B (- 5, 7), then scalar component of AB are
x2 – x1 = – 5 – 2 = – 7and y2 – y1 = 7 – 1 = 6.

Question 24.
For what values of a⃗ , the vectors 2î – 3ĵ + 4k̂ and aî + 6ĵ – 8k̂ are collinear? (Delhi 2011)
Answer:
If a⃗  and b⃗  are collinear, then use the condition a⃗  = λb⃗ , where λ is some scalar.

Let given vectors are a⃗  = 2î – 3ĵ + 4k̂ and a⃗  = aî + 6ĵ – 8k̂
We know that, vectors a⃗  and b⃗  are said to be collinear, if
a⃗  = k. b⃗ , where k is a scalar.
∴ 2î – 3ĵ + 4k̂ = k(aî + 6ĵ – 8k̂)

On comparing the coefficients of î and ĵ, we get
2 = ka and -3 = 6k ⇒ k = –12
∴ 2 = –12a ⇒ a = -4

Question 25.
Write the direction cosines of vector -2î + ĵ – 5k̂. (Delhi 2011)
Answer:
Direction cosines of the vector aî + bĵ + ck̂ are
Vector Algebra Class 12 Maths Important Questions Chapter 10 14

Question 26.
Write the position vector of mid-point of the vector joining points P(2, 3, 4) and Q (4, 1, – 2). (Foreign 2011)
Answer:
Mid-point of the position vectors
a⃗  = a1î + a2ĵ + a3k̂ and
b⃗  = b1î + b2ĵ + b3k̂ is a⃗ +b⃗ 2 or (a1+b1)i^+(a2+b2)j^+(a3+b3)k^2

Given points are P(2, 3, 4) and Q(4,1,-2) whose position vectors are OP = 2 î + 5ĵ + 4k̂ and OQ =4î + ĵ – 2k̂.
Now, position vector of mid-point of vector joining points P(2, 3, 4) and Q(4, 1, – 2) is
Vector Algebra Class 12 Maths Important Questions Chapter 10 15

Question 27.
Write a unit vector in the direction of vector a⃗  = 2î + ĵ + 2k̂. (All India 2011; Delhi 2009)
Answer:
We know that, unit vector in the direction of â is â = a⃗ |a⃗ |

Required unit vector in the direction of vector
a⃗  = 2î + ĵ + 2k̂
Vector Algebra Class 12 Maths Important Questions Chapter 10 16

Question 28.
Find the magnitude of the vector a⃗  = 3î – 2ĵ + 6k̂. (All India 2011C: Delhi 2008)
Answer:
Magnitude of a vector r = xî + yĵ + zk̂ is |r⃗ | = x2+y2+z2

Given vector is a = 3i – 2/ + 6fc.
∴ Magnitude of a⃗  = |a⃗ |=(3)2+(2)2+(6)2
9+4+36=49 = 7 units

Question 29.
Find a unit vector in the direction of vector a⃗  = 2î + 3ĵ + 6k̂. (All India 2011C)
Answer:
27i^+37j^+67k^

Question 30.
If A, B and C are the vertices of a ΔABC, then what is the value of AB+BC+CA ? (Delhi 2011C)
Answer:
Let ΔABC be the given triangle.
Vector Algebra Class 12 Maths Important Questions Chapter 10 17
Now, by triangle law of vector addition,
Vector Algebra Class 12 Maths Important Questions Chapter 10 18

Question 31.
Find a unit vector in the direction of a⃗  = 2î – 3ĵ + 6k̂. (Delhi 2011c)
Answer:
27i^37j^+67k^

Question 32.
Find a vector in the direction of a⃗  = 2î – ĵ + 2k̂, which has magnitude 6 units. (Delhi 2010C)
Answer:
4î – 2ĵ + 4k̂

Question 33.
Find the position vector of mid-point of the line segment AB, where A is point (3, 4, -2) and Bis point (1, 2, 4). (Delhi 2010)
Answer:
2î + 3ĵ + k̂

Question 34.
Write a vector of magnitude 9 units in the direction of vector -2î + ĵ + 2k̂. (All India 2010)
Answer:
-6î + 3ĵ + 6k̂

Question 35.
Write a vector of magnitude 15 units in the direction of vector î – 2ĵ + 2k̂. (Delhi 2010)
Answer:
5î – 10ĵ + 10k̂

Question 36.
What is the cosine of angle which the vector √2î + ĵ + k̂ makes with Y-axis? (Delhi 2010)
Answer:
12

Question 37.
Find a vector of magnitude 5 units and parallel to the resultant of a⃗  = 2î + 3ĵ – k̂ and b⃗  = î – 2ĵ + k̂. (Delhi 2011)
Answer:
First, find resultant of the vectors a and o, which is a⃗  + b⃗ . Then, find a unit vector in the direction of a⃗  + b⃗ . After this, the unit vector is multiplying by 5.

Given, a⃗  = 2î + 3ĵ – k̂ and b⃗  = î – 2ĵ + k̂.
Now, resultant of above vectors = a⃗  + b⃗ 
= (2î + 3ĵ – k̂) + (î – 2ĵ + k̂) = 3î + ĵ
Vector Algebra Class 12 Maths Important Questions Chapter 10 19

Question 38.
Let a⃗  = î + ĵ + k̂, b⃗  = 4î – 2ĵ + 8k̂ and c⃗  = î – 2ĵ + k̂. Find a vector of magnitude 6 units, which is parallel to the vector 2a⃗  – b⃗  + 8 c⃗ . (All India 2010)
Answer:
First, find the vector 2 a – b + 3c, then find a unit vector in the direction of 2a-b + 3c.
After this, the unit vector is multiplying by 6.
Given, a⃗  = î + ĵ + k̂, b⃗  = 4î – 2ĵ + 8k̂ and c⃗  = î – 2ĵ + k̂
∴ 2a⃗ b⃗ +3c⃗ 
= 2 (î + ĵ + k̂) – (4î – 2ĵ + 3k̂) + 3 (î – 2ĵ + k̂)
= 2î + 2ĵ + 2k̂ – 4î + 2ĵ – 3k̂ + 3î – 6ĵ + 3k̂
⇒ 2a⃗ b⃗ +3c⃗  = î – 2ĵ + 2k̂

Now, a unit vector in the direction of vector
Vector Algebra Class 12 Maths Important Questions Chapter 10 20
Hence, vector of magnitude 6 units parallel to the Vector 2a⃗ b⃗ +3c⃗  = 6(13i^23j^+23k^)
= 2î – 4ĵ + 4k̂

Question 39.
Find the position vector of a point R, which divides the line joining two points P and Q whose position vectors are 2a⃗  + b⃗  and a⃗  – 8b⃗  respectively, externally in the ratio 1 : 2. Also, show that P is the mid-point of line segment RO. (Delhi 2010)
Answer:
Given, OP = Position vector of P = 2a⃗  + b⃗ 
and OQ = Position vector of Q = a⃗  – 3b⃗ 

Let OR be the position vector of point R, which divides PQ in the ratio 1 : 2 externally
Vector Algebra Class 12 Maths Important Questions Chapter 10 21

Now, we have to show that P is the mid-point of RQ,
Vector Algebra Class 12 Maths Important Questions Chapter 10 22
Hence, P is the mid-point of line segment RQ.

Product of Two Vectors and Scalar Triple Product

Question 1.
Find the magnitude of each of the two vectors a⃗  and b⃗ , having the same magnitude such that the angle between them is 60° and their scalar product is 92. (CBSE 2018)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 23

Question 2.
Find the value of [î, k̂, ĵ], (CBSE 2018C)
Answer:
[î, k̂, ĵ] = î ∙ (k̂× ĵ)
= -[[î, k̂, ĵ] = –100010001 = – 1

Question 3.
Find λ and μ, if (î + 3ĵ + 9k̂) × (3î – λĵ + μk̂) = 0. (All India 2016)
Answer:
Given, (î + 3ĵ + 9k̂) × (3î – λĵ + μk̂)
Vector Algebra Class 12 Maths Important Questions Chapter 10 24
= î(3μ + 9λ) ĵ k̂
On comparing the coefficients of î, ĵ and k̂ , we get
3μ + 9λ = 0, – μ + 27 = 0 and – λ – 9 = 0
⇒ μ = 27 and – λ = 9
⇒ μ = 27 and λ = – 9
Also, the values of μ and λ satisfy the equation
3μ + 9λ = 0.
Hence, μ = 27 and λ = – 9.

Question 4.
Write the number of vectors of unit length perpendicular to both the vectors a⃗  = 2î + ĵ + 2k̂ and b⃗  = ĵ + k̂. (All India 2016)
Answer:
We know that, unit vectors perpendicular to a⃗ 
and b⃗  are ±(a⃗ ×b⃗ |a⃗ ×b⃗ |)
So, there arc two unit vectors perpendicular to the given vectors.

Question 5.
If a⃗ ,b⃗ ,c⃗  are unit vectors such that a⃗ +b⃗ +c⃗ =0 = 0, then write the value of a⃗ b⃗ +b⃗ c⃗ +c⃗ a⃗ . (Foreign 2016)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 25

Question 6.
If |a⃗ ×b⃗ |2+|a⃗ b⃗ |2 = 400 and |a⃗ | = 5, then write the value of |b⃗ |. (Foreign 2016)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 26

Question 7.
Find λ, if the vectors a⃗  = î + 3ĵ + k̂, b⃗  = 2 î – ĵ – k̂ and c⃗  = λĵ + 3 k̂ are coplanar. (Delhi 2015)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 27
⇒ 1(- 3 + λ) – 3(6) + 1(2λ) = 0
[expanding along R1]
⇒ – 3 + λ – 18 + 2λ = 0
⇒ 3λ = 21
∴ λ = 7

Question 8.
If a⃗  = 7î + ĵ – 4k̂ and b⃗  = 2î + 6ĵ + 3k̂, then find the projection of a⃗  on b⃗ . (Delhi 2015)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 28

Question 9.
If â, b̂ and ĉ are mutually perpendicular unit vectors, then find the value of |2â + b̂ + ĉ |. (All India 2015)
Answer:
Given â, b̂ and ĉ are mutually perpendicular unit vectors, i.e.
Vector Algebra Class 12 Maths Important Questions Chapter 10 29

Question 10.
Write a unit vector perpendicular to both the vectors a⃗  = î + ĵ + k̂ and b⃗  = î + ĵ. (All India 2015)
Answer:
First, determine perpendicular vectors of a⃗  and b⃗ , i.e., a⃗ ×b⃗ . Further , determine perpendicular unit vector by using formula a⃗ ×b⃗ |a⃗ ×b⃗ |.

Given vector are a⃗  = î + ĵ + k̂ and b⃗  = î + ĵ
As we know the, vectors a⃗ ×b⃗  is perpendicular to both the vectors, so let us first evaluate a⃗ ×b⃗ .
Then, a⃗ ×b⃗  = i^11j^11k^10
= î(0 -1) – ĵ(0 – 1) + k̂(1 – 1)
= – î + ĵ
Then , the unit vector perpendicular to both a⃗  and b⃗  is given by
Vector Algebra Class 12 Maths Important Questions Chapter 10 30

Question 11.
Find the area of a parallelogram whose adjacent sides are represented by the vectors 2 î – 3 k̂ and 4 ĵ + 2 k̂. (Foreign 2015)
Answer:
Let adjacent sides of a parallelogram bc
a⃗  = 2 î – 3 k̂ and b⃗  = 4 ĵ + 2 k̂.
Vector Algebra Class 12 Maths Important Questions Chapter 10 31

Question 12.
If a⃗  and b⃗  are perpendicular vectors, |a⃗  + b⃗ | = 13 and |a⃗ | = 5, then find the value of |b⃗ |. (All India 2014)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 32

Question 13.
If a⃗  and b⃗  are two unit vectors such that a⃗  + b⃗  is also a unit vector, then find the angle between a⃗  and b⃗ . (Delhi 2014)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 33

Question 14.
Find the projection of the vector î + 3ĵ + 7k̂ on the vector 2î – 3 ĵ + 6k̂. (Delhi 2014)
Answer:
let a⃗  = î + 3ĵ + 7k̂ and a⃗  = 2î – 3 ĵ + 6k̂

Question 15.
Write the projection of vector î + ĵ + k̂ along the vector ĵ. (Foreign 2014)
Answer:
1

Question 16.
Write the value of the following. î × (ĵ + k̂) + ĵ × (k̂ + î) + k̂ × (î + ĵ). (Foreign 2014)
Answer:
we have, î × (ĵ + k̂) + ĵ × (k̂ + î) + k̂ × (î + ĵ)
= î × ĵ + î × k̂ × ĵ × k̂ + ĵ × î + k̂ × î + k̂ × ĵ
[∵ cross product is distributive over addition]
= k̂ – ĵ + î – k̂ + ĵ – î = 0⃗ 
[∵ î × ĵ = k̂, î × k̂ = – ĵ, ĵ × k̂ = î, ĵ × î = – k̂, k̂ × î = ĵ, k̂ × ĵ = – î ]

Question 17.
If vectors a⃗  and b⃗  are such that |a⃗ | = 3, |b⃗ | = 2/3 and a⃗  × b⃗  is a unit vector, then write the angle between a⃗  and b⃗ . (Delhi 2014: All India 2010)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 34

Question 18.
Find a⃗ (b⃗ ×c⃗ ), if a⃗  = 2î + ĵ + 3k̂, b⃗  = -î + 2ĵ + k̂, and c⃗  = 3î + ĵ + 2k̂. (All India 2014)
Answer:
Given, a⃗  = 2î + ĵ + 3k̂, b⃗  = -î + 2ĵ + k̂, and c⃗  = 3î + ĵ + 2k̂.
Vector Algebra Class 12 Maths Important Questions Chapter 10 35
= 2(4 – 1) – 1 (- 2 – 3) + 3( – 1 – 6)
= 2 × 3 – 1 × (-5) + 3 × (- 7)
= 6 + 5 – 21 = 11 – 21 = – 10

Question 19.
If a⃗  and b⃗  are unit vectors, then find the angle between a⃗  and b⃗ , given that (√3a⃗  – b⃗ ) is a unit vector. (Delhi 2014C)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 36

Question 20.
If |a⃗ | = 8, |b⃗ | = 3 and||a⃗ ×b⃗ || = 12, find the angle between a⃗  and b⃗ . (All India 2014C)
Answer:
let θ be the angle between latex]\vec{a}[/latex] and b⃗ .
Vector Algebra Class 12 Maths Important Questions Chapter 10 37

Question 21.
Write the projection of the vector a⃗  = 2 î – ĵ + k̂ on the vector b⃗  = î + 2ĵ + 2k̂. (Delhi 2014 C)
Answer:
23

Question 22.
Write the value of λ, so that the vectors a = 2î + λĵ + k̂ and b = î – 2ĵ + 3k̂ are perpendicular to each other. (Delhi 2013C, 2008)
Answer:
Given vectors are a⃗  = 2î + λĵ + k̂
and b⃗  = î – 2ĵ + 3k̂
Since, vectors are perpendicular.
∴ a⃗ b⃗  = 0
⇒ (2î + λĵ + k̂) ∙ (î – 2ĵ + 3k̂)
⇒ 2 – 2λ + 3 = 0
∴ λ = 5/2

Question 23.
Write the projection of (b⃗  + c⃗ ) on a⃗ , where a⃗  = 2î – 2ĵ + k̂, b⃗  = î + 2ĵ – 2k̂ and c⃗  = 2î – ĵ + 4k̂. (All India 2013 C)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 38

Question 24.
Write the projection of the vector 7î + ĵ – 4k̂ on the vector 2î + 6 ĵ + 3k̂. (Delhi 2013C)
Answer:
87

Question 25.
If a⃗  and b⃗  are two vectors such that |a⃗  + b⃗ | = |a⃗ |, then prove that vector 2a⃗  + b⃗  is perpendicular to vector b. (Delhi 2013)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 39

Question 26.
Find |x⃗ |, if for â unit vector a, (x⃗ a⃗ )(x⃗ +a⃗ ) = 15. (All India 2013)
Answer:
Given, a⃗  is a unit vector. Then, |a⃗ | = 1.
Vector Algebra Class 12 Maths Important Questions Chapter 10 40

Question 27.
Find λ, when projection of a⃗  = λî + ĵ + 4k̂ on b⃗  = 2 î + 6 ĵ + 3k̂ is 4 units. (Delhi 2012)
Answer:
Given, a⃗  = λî + ĵ + 4k̂ on b⃗  = 2 î + 6 ĵ + 3k̂ and projection of a⃗  and b⃗  = 4.
Vector Algebra Class 12 Maths Important Questions Chapter 10 41
⇒ 2λ + 18 = 28
⇒ 2λ = 10
∴ λ = 5

Question 28.
Write the value of (k̂ × ĵ) . î + ĵ . k̂. (All India 2012)
Answer:
Use the results k̂ × ĵ = – î
ĵ ∙ k̂ and î ∙ î = 1 and simplify it.

Given, (k̂ × î) ∙ î + ĵ ∙ k̂ = (- î) ∙ î + ĵ ∙ k̂
= – (î ∙ î) + 0 = – 1 [∵ (î ∙ î) = 1]

Question 29.
If a⃗ a⃗  = 0 and a⃗ b⃗  = 0, then what can be concluded about the vector b⃗ ? (Foreign 2011)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 42
From Eqs. (i) and (ii). it may be concluded that b⃗  is either zero or non-zero perpendicular vector.

Question 30.
Write the projection of vector î – ĵ on the vector î + ĵ. (All India 2011)
Answer:
0

Question 31.
Write the angle between vectors a⃗  and b⃗  with magnitudes √3 and 2 respectively, having a⃗ b⃗  = √6. (All India 2011)
Answer:
let θ be the angle between a⃗  and b⃗ , then use the following formula
cos θ = a⃗ b⃗ |a⃗ ||b⃗ |.

Vector Algebra Class 12 Maths Important Questions Chapter 10 43

Question 32.
For what value of λ are the vectors î + 2λĵ + k̂ and 2î + ĵ – 3k̂ perpendicular? (All India 2011C)
Answer:
12

Question 33.
If |a⃗ | = √3, |b⃗ | = 2 and angle between a⃗  and b⃗  is 60°, then find a⃗ .b⃗ . (Delhi 2011C)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 44

Question 34.
Find the value of λ, if the vectors 2î + λĵ + 3k and 3î + 2ĵ – 4k̂ are perpendicular to each other. (All India 2010c)
Answer:
3

Question 35.
If |a⃗ | = 2, |b⃗ | = 3 and a⃗ .b⃗  = 3, then find the projection of b⃗  on a⃗ . (All India 2010C)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 45

Question 36.
If a⃗  and b⃗  are two vectors, such that |a⃗ b⃗ |=|a⃗ ×b⃗ |, then find the angle between a⃗  and b⃗ . (All India 2010)
Answer:
Use the following formulae:
a⃗ b⃗  = |a⃗ ||b⃗ | cos θ
and |a⃗ ×b⃗ | = |a⃗ ||b⃗ | sin θ
where, θ is the angle between a⃗  and b⃗ .
Vector Algebra Class 12 Maths Important Questions Chapter 10 46

Question 37.
Find λ, if (2î + 6ĵ + 14k̂) × (î – λĵ + Ik̂) = 0. (All India 2010)
Answer:
– 3

Question 38.
If the sum of two unit vectors a and b is a unit vector, show that the magnitude of their difference is √3. (Delhi 2019, 2012c)
Answer:
let c⃗  = a⃗  + b⃗ . Then, according to given condition c⃗  is a unit vector, i.e. |c⃗ | = 1.
Vector Algebra Class 12 Maths Important Questions Chapter 10 47
[taking positive square root, as magnitude cannot be negative]

Question 39.
If a⃗  = 2î + 5ĵ + k̂, b⃗  = î – 2 ĵ + k̂ and c⃗  = – 3î + ĵ + 2k̂, find [a⃗ b⃗ c⃗ ]. (Delhi 2019)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 48
= 2(- 4 – 1) – 3(2 + 3) + 1(1 – 6)
= – 10 – 15 – 5 = – 30

Question 40.
If |a⃗ | = 2, |b⃗ | = 7 and a⃗ ×b⃗  = 3î + 2ĵ + 6k̂, find the angle between a⃗  and b⃗ . (All India 2019)
Answer:
let θ be the angle between a⃗  and b⃗ .
Vector Algebra Class 12 Maths Important Questions Chapter 10 49

Question 41.
Find the volume of cuboid whose edges are given by -5î + 7ĵ + 5k̂, -5î + 7ĵ – 5k̂ and 7î – 5 ĵ – 5k̂. (All India 2019)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 50
= |- 3 (- 21 – 15) – 7 (15 + 21) + 5(25 – 49)|
= |1108 – 252 – 120|
= 264 cubic units

Question 42.
Show that the points A(-2î + 5ĵ + 5k̂), B(î + 2 ĵ + 5k̂) and C(7î – k̂) are collinear. (All India 2019)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 51

Question 43.
Find |a⃗ ×b⃗ |, if a⃗  = 2î + ĵ + 5k̂ and b⃗  = 3î + 5ĵ – 2k̂. (All India 2019)
Answer:
We have, a⃗  = 2î + ĵ + 3k̂ and b⃗  = 3î + 5ĵ – 2k̂
∴ a⃗ ×b⃗  = i^23j^15k^32
= î ( – 2 – 15) – ĵ (- 4 – 9) + k̂(10 – 3)
= – 17î + 13ĵ + 7k̂

Question 44.
If θ is the angle between two vectors î – 2 ĵ + 3k̂ and 3î – 2 ĵ + k̂, find sin θ. (CBSE 2018)
Answer:
let a⃗  = î – 2 ĵ + 3k̂ and b⃗  3î – 2 ĵ + k̂
Vector Algebra Class 12 Maths Important Questions Chapter 10 52

Question 45.
If a⃗ +b⃗ +c⃗  = 0 and |a⃗ | = 5, |b⃗ | = 6 and |c⃗ | = 9, then find the angle between a⃗  and b⃗ . (CBSE 2018C)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 53
(5)2 + 2 × 5 × 6 × cos θ + (6)2 = (9)2
⇒ 25 + 60 cos θ + 36 = 81
⇒ cos θ = 2060=13
⇒ θ = cos-1(13)

Question 46.
If î + ĵ + k̂, 2î + 5ĵ, 5î + 2ĵ – 5k̂ and î – 6ĵ – k̂ respectively, are the position vectors of points A, B, C and D, then find the angle between the straight lines AB and CD. Find whether AB and CD are collinear or not. (Delhi 2019)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 54

Question 47.
The scalar product of the vector a⃗  = î + ĵ + k̂ with a unit vector along the sum of the vectors b⃗  = 2î + 4ĵ – 5k̂ and c⃗  = λî + 2ĵ + 5k̂ is equal to 1. Find the value of λ and hence find the unit vector along b⃗  + c⃗ . (All India 2019)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 55

Question 48.
Let a⃗  = 4 î + 5ĵ – k̂, b⃗  = î – 4ĵ + 5k̂ and c⃗  = 3î + ĵ – k̂. Find a vector which is perpendicular to both c⃗  and b⃗  and d⃗ a⃗  = 21. (CBSE 2018)
Answer:
We have, a⃗  = 4 î + 5ĵ – k̂, b⃗  = î – 4ĵ + 5k̂ and c⃗  = 3î + ĵ – k̂.
Since, d⃗  is perpendicular to both c⃗  and b⃗ .
Vector Algebra Class 12 Maths Important Questions Chapter 10 56
= λ[î(5 – 4) – ĵ(15 + 1) + k̂(- 12 – 1)]
= λ(î – 16ĵ – 13k̂)
Also, it is given that a⃗ a⃗  = 21
∴ λ(î – 16ĵ – 13k̂) ∙ (4î + 5ĵ – k̂) = 21
⇒ λ(4 – 80 + 13) = 21
⇒ λ(- 63) = 21
⇒ λ = 13
Now from Eq. (j), we get
d⃗  = –13(î – 16ĵ – 13k̂)

Question 49.
Find x such that the four points A(4, 4, 4), B(5, x, 8), C(5, 4, 1) and D (7, 7, 2) are coplanar. (CBSE 2018C)
Answer:
Given points are A(4, 4, 4), B (5, x, 8), C(5, 4, 1) and D(7, 7, 2), then position vectors of A, B, C and D respectively, are
Vector Algebra Class 12 Maths Important Questions Chapter 10 57
⇒ 1(0 + 9) – (x – 4) (- 2 + 9) + 4(3 – 0) = 0
⇒ 9 – (x – 4) (7) + 12 = 0
⇒ 9 – 7x + 28 + 12 = 0
⇒ 49 – 7x = 0
⇒ 7x = 49
⇒ x = 7

Question 50.
Find the value of x such that the points A(3, 2, 1), B(4, x, 5), C(4, 2,- 2) and D (6, 5, -1) are coplanar. (All India 2017)
Answer:
5

Question 51.
If a⃗ b⃗  and c⃗  are three mutually perpendicular vectors of the same magnitude, then prove that a⃗ +b⃗ +c⃗  is equally inclined with the vectors a⃗ b⃗  and c⃗ . (Delhi 2017, 2013C, 2011)
Answer:
If three vectors a⃗ b⃗  and c⃗  are mutually perpendicular to each other, then a⃗ b⃗ =b⃗ c⃗  = c⃗ a⃗  = 0 and if all three vectors a⃗ b⃗  and c⃗  are equally inclined with the vector (a⃗ +b⃗ +c⃗ ) that means each vector a⃗ b⃗  and c⃗  makes equal angle with (a⃗ +b⃗ +c⃗ ) by using formula
cos θ = a⃗ b⃗ |a⃗ ||b⃗ |.
Vector Algebra Class 12 Maths Important Questions Chapter 10 58

Question 52.
Using vectors, find the area of the ΔABC, whose vertices are A(1, 2, 5), 5(2, -1, 4) and C(4, 5, -1). (Delhi 2017; All India 2013)
Answer:
Let the position vectors of the verices A, B and C of ΔABC be
Vector Algebra Class 12 Maths Important Questions Chapter 10 59

Question 53.
Let a⃗  = î + ĵ + k̂, b⃗  = î + 0 ∙ ĵ + 0 ∙ k̂ and c⃗  = c1î + c2ĵ + c3k̂, then
(a) Let c1 = 1 and c2 = 2, find c3 which makes a⃗ b⃗  and c⃗  coplanar.
(b) If c2 = – 1 and c3 = 1, show that no value of c1 can make a⃗ b⃗  and c⃗  coplanar. (Delhi 2017)
Answer:
Given, a⃗  = î + ĵ + k̂, b⃗  = î + 0 ∙ ĵ + 0 ∙ k̂ and c⃗  = c1î + c2ĵ + c3
The given vectors are coplanar iff [a⃗ b⃗ c⃗ ] = 0

(a) If c1 = 1 and c2 = 2,
Then, from Eq.(i), we get
11110210c3 = 0
Vector Algebra Class 12 Maths Important Questions Chapter 10 60
⇒ – 1(c3 – 0) + 1(2 – 0) = 0
⇒ – c3 + 2 = 0
⇒ – c3 = – 2
⇒ c3 = 2

(b) If c2 = – 1 and c3 = 1, then from Eq. (i), we get
11c1101101 = 0
⇒ 1(0) – 1(1 – 0) + 1(- 1 – 0) = 0
⇒ 0 – 1 – 1 = 0
⇒ – 2 ≠ 0
∴ No value of c1 can make 1’ and coplanar.
Hence proved

Question 54.
Show that the points A, B, C with position vectors 2î – ĵ + k̂, î – 5ĵ – 5k̂ and 5î – 4ĵ – 4k̂ respectively, are the vertices of a right-angled triangle. Hence find the area of the triangle. (All India 2017)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 61

Question 55.
Show that the vectors a⃗ b⃗  and c⃗  are coplanar, if a + b, 6+ c and c+ a are coplanar. (Delhi 2016, Foreign 2014)
Or
Prove that, for any three vectors a⃗ b⃗  and c⃗ ,[a⃗ +b⃗ b⃗ +c⃗ c⃗ +a⃗ ]=2[a⃗ b⃗ c⃗ ]. (Delhi 2014)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 62

Question 56.
Show that the four points A (4, 5, 1), B(0, -1, -1), C(3, 9, 4) and D (-4, 4, 4) are coplanar. (All India 2016)
Or
Show that the four points A, B, C and D with position vectors 4î + 5ĵ + k̂, – ĵ – k̂, 3î + 9ĵ + 4k̂ and 4(- î + ĵ + k̂), respectively are coplanar. (All India 2014)
Answer:
Let the position vector of points A, B, C and D are
Vector Algebra Class 12 Maths Important Questions Chapter 10 63
= – 4(12 + 3) + 6 (- 3 + 24) – 2(1 + 32)
= – 60 + 126 – 66 = 0
Hence, the four points A, B, C and D are coplanar.

Question 57.
The two adjacent sides of a parallelogram are 2î – 4ĵ – 5k̂ and 2î + 2 ĵ + 3k̂. Find the two unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of the parallelogram. (All India 2016)
Answer:
Let ABCD be the given parallelogram with
Vector Algebra Class 12 Maths Important Questions Chapter 10 64

Question 58.
If a⃗ ×b⃗ =c⃗ ×d⃗  and a⃗ ×c⃗ =b⃗ ×d⃗ , then show that a⃗ d⃗  is parallel to b⃗ c⃗ , where a⃗ d⃗  and b⃗ c⃗ . (Foreign 2016; Delhi 2009)
Answer:
Use the result, if two vectors are parallel, then their cross-product will be a zero vector.
Vector Algebra Class 12 Maths Important Questions Chapter 10 65

Question 59.
If r⃗  = xî + yĵ + zk̂, find (r⃗ ×i^)(r⃗ ×j^) + xy. (Delhi 2015)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 66

Question 60.
If a⃗  = î + 2ĵ + k̂, b⃗  = 2î + ĵ and c⃗  = 3î – 4 ĵ – 5k̂, then find a unit vector perpendicular to both of the vectors (a⃗ b⃗ ) and (c⃗ b⃗ ). (All India 2015)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 67

Question 61.
Find the value of λ so that the four points A, B,C and D with position vectors 4 î + 5ĵ + k̂, -ĵ – k̂,3i + Xj+4k and – 4 î + 4ĵ + 4 k̂, respectively are coplanar. (Delhi 2015C)
Answer:
Use the condition that four points with position vectors A⃗ ,B⃗ ,C⃗  and D⃗  are coplanar, if
[AB,AC,AD]=0 = 0.
Vector Algebra Class 12 Maths Important Questions Chapter 10 68
On expanding along R1, we get
⇒ – 4(3λ – 15 + 3) + 6(- 3 + 24) – 2(1 + 8λ – 40) = 0
⇒ – 4(3λ – 12) + 6(21) – 2(8λ – 39) = 0
⇒ – 12λ + 48 + 126 – 16λ + 78 = 0
⇒ – 28λ + 252 = 0
λ = 9

Question 62.
Prove that [a⃗ b⃗ +c⃗ d⃗ ]=[a⃗ b⃗ d⃗ ]+[a⃗ c⃗ d⃗ ]. (All India 2015C)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 69

Question 63.
If a⃗  = 2î – 3ĵ + k̂, b⃗  = – î + k̂, c⃗  = 2 ĵ – k̂ are three vectors, find the area of the parallelogram having diagonals  and . (Delhi 2014C)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 70

Question 64.
Vectors a⃗ ,b⃗  and c⃗  are such that a⃗ +b⃗ +c⃗ =0 and |a⃗ | = 3, |b⃗ | = 5 and |c⃗ | = 7. Find the angle between a⃗  and b⃗ . (Delhi 2014,2008; All India 2008)
Answer:
Ï€3

Question 65.
The scalar product of the vector a⃗  = î + ĵ + k̂ with a unit vector along the sum of vectors b⃗  = 2î + 4ĵ – 5k̂ and c⃗  = λî + 2ĵ + 3k̂ is equal to one. Find the value of λ and hence, find the unit vector along b⃗  + c⃗ . (All India 2014)
Or
The scalar product of vector i + j + k with the unit vector along the sum of vectors 2î + 4ĵ – 5k̂ and λî + 2ĵ + 3k̂ is equal to one. Find the value of λ. (All India 2009,2008C)
Answer:
First, determine the unit vector of b⃗ +c⃗ , i.e. b⃗ +c⃗ |b⃗ +c⃗ |. Further put a⃗ (b⃗ +c⃗ )|b⃗ +c⃗ | = 1 and then determine the value of λ.
Vector Algebra Class 12 Maths Important Questions Chapter 10 71
⇒ (λ + 6)2 = λ2 + 4λ + 44 [squaring both sides]
⇒ λ2 + 36 + 12λ + 4λ + 44
⇒ 8λ = 8
⇒ λ = 1
Hence, the value of λ is 1.
On substituting the value of λ in Eq. (1), we get Unit vector along b⃗ +c⃗ 
Vector Algebra Class 12 Maths Important Questions Chapter 10 72

Question 66.
Find the vector p⃗  which is perpendicular to both Î±⃗  = 4î + 5ĵ – k̂ and Î²⃗  = î – 4ĵ + 5k̂ and p⃗ q⃗  = 21, where q⃗  = 3i + j – k. (All India 2014C)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 73
= î (25 – 4) – ĵ (20 + 1) + k̂(- 16 – 5)
= î(21) – ĵ(21) + k̂(- 21)
= 21î – 21ĵ – 21k̂
So, p⃗  = 21λk̂—21λ?—21λk [fromEq.(i)] ….. (ii)
Also, given that p⃗ q⃗  = 21
∴ (21λî – 21λĵ – 21λk̂) . (3î + ĵ – k̂) = 21
⇒ 63λ – 21λ + 21λ = 21
⇒ 63λ = 21
⇒ λ = 13
On putting λ = 13 in Eq. (ii), we get
Vector Algebra Class 12 Maths Important Questions Chapter 10 74
which is the required vector.

Question 67.
Find the unit vector perpendicular to both of the vectors a⃗ +b⃗  and a⃗ b⃗  where, a⃗  = î + ĵ + k̂ and b⃗  = î + 2ĵ + 3k̂. (Foreign 2014)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 75
⇒ 2x + 3y + 4z = 0 …… (ii)
and (xî + yĵ + zk̂) . (- ĵ – 2k̂) = 0
⇒ – y – 2z = 0
⇒ y = – 2z
On putting the value of yin Eq. (ii), we get
2x + 3 (- 2z) + 4z = 0
⇒ x = z
On substituting the value of x and y in Eq. (1),
we get
⇒ z2 + 4z2 + z2 = 1
⇒ 6z2 = 1
⇒ z = ± 16
then, x = ± 16
and y = ± 26
Hence, the required vectors are
Vector Algebra Class 12 Maths Important Questions Chapter 10 76

Question 68.
Find the unit vector perpendicular to the plane ABC where the position vectors of A, B and C are 2î – ĵ + k̂, î + ĵ + 2k̂ and 2î + 3k̂, respectively. (All India 2014C)
Answer:
A unit vector perpendicular to plane ABC is
AB×AC|AB×AC|

Let O be the origin of reference.
Vector Algebra Class 12 Maths Important Questions Chapter 10 77

Question 69.
Dot product of a vector with vectors î – ĵ + k̂, 2î + ĵ – 3k̂ and î + ĵ + k̂ are respectively 4, 0 and 2. Find the vector. (Delhi 2013C)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 78
⇒ a1 + a2 + a3 = 2
On subtracting Eq. (iii) from Eq. (i), we get
– 2a2 = 2
⇒ a2 = – 1
On substituting a2 = – 1 in Eq. (ii) and (iii),
we get
2a2 – 3a3 = 1 …… (iv)
⇒ a1 + a3 = 3
On multiplying Eq. (v) by 3 and then adding with Eq. (iv), we get
5a1 = 1 + 9 = 10
⇒ a1 = 2
On substituting a1 = 2 in Eq. (v), we get
a3 = 1
Hence, the vector is a⃗  = 2î – ĵ + k̂

Question 70.
Find the values of λ for which the angle between the vectors a⃗  = 2λ2î + 4λĵ + k̂ and b⃗  = 7î – 2ĵ + λk̂ is obtuse. (All India 2013C)
Answer:
let θ be the obtuse angle between the vectors
Vector Algebra Class 12 Maths Important Questions Chapter 10 79
14λ2 – 7λ < 0
= 2λ2 – λ < 0
Either λ < 0, 2λ – 1 > 0 or λ > 0, 2λ – 1 < 0
= Either λ < 0, λ > 12 or λ > 0, λ < 12 Clearly, first option is impossible. ∴ λ > 0, λ < 12
0 < λ < 12
λ ∈ (0,12)

Question 71.
If a, b and c are three vectors such that each one is perpendicular to the vector obtained by sum of the other two and |a⃗ | = 3, |b⃗ | = 4 and |c⃗ | = 5, then prove that |a⃗ +b⃗ +c⃗ | = 5√2. (All India 2013C, 2010C)
Or
If a⃗ ,b⃗  and c⃗  are three vectors, such that |a⃗ | = 3, |b⃗ | = 4 and |c⃗ | = 5 and each one of these is perpendicular to the sum of other two, then find |a⃗ +b⃗ +c⃗ |. (All India 2011C, 2010C)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 80

Question 72.
If a⃗  = 3î – ĵ and b⃗  = 2î + ĵ – 3k̂, then express b⃗  in the form b⃗ =b⃗ 1+b⃗ 2, where b⃗ 1a⃗  and b⃗ 2a⃗ . (All India 2013C)
Answer:
Given a⃗  = 3î – ĵ and b⃗  = 2î + ĵ – 3k̂
Let b1 = x1î + y1ĵ + z1k̂ are two vectors such that b1+b2=b⃗ ,b1a⃗  and b2a⃗ 

Consider, b⃗ 1+b⃗ 2=b⃗ 
⇒ (x1 + x2)î + (y1 + y2)ĵ + (z1 + z2)k̂ = 2î + ĵ – 3k̂

On comparing the coefficient of î ĵ and k̂ both sides, we get
x1 + x2 = 2
y1 + y2 = 1
z1 + z2 = -3

Now, consider b1a⃗ 
⇒ x13=y11=z10
⇒ x1 = 3λ, y1 = -λ,and z1 = 0 …(iv)

On substituting the values of x, y and z, from Eq. (iv) to Eq. (i), (ii) and (iii), respectively, we get
x2 = 2- 3λ, y2 = -1 + λ and z2 = -3 …(v)
Since, b2 ± a , therefore b2 a = 0
⇒ 3x2 – y2 = 0
⇒ 3 (2 – 3λ) – (1 + λ) = 0
⇒ 6 – 9λ – 1 – λ = 0
⇒ 5 – 10λ = 0
⇒ λ = 12

On substituting λ = 12 in Eqs. (iv) and (v), we get
Vector Algebra Class 12 Maths Important Questions Chapter 10 81

Question 73.
If a⃗  = î + ĵ + k̂ and b⃗  = ĵ – k̂, then find a vector c⃗ , such that a⃗ ×c⃗ =b⃗  and a⃗ c⃗  = 3. (Delhi 2013, 2008)
Answer:
Given a⃗  = î + ĵ + k̂ and b⃗  = ĵ – k̂
Let c⃗  = xî + yĵ + zk̂
Vector Algebra Class 12 Maths Important Questions Chapter 10 82
= î (z – y) – ĵ(z – x) + k̂(y – x)
Now, a⃗ ×c⃗ =b⃗  [given]
= î(z – y) + ĵ(x – z) + k̂(y – x)
= 0î + 1ĵ + (-1)k̂ [∵ b⃗  = ĵ – k̂]

On comparing the coefficients from both sides, we get
z – y = 0,x – z = 1, y – x = -1
⇒ y = z and x – y = 1…(i)

Also given, a⃗ c⃗  = 3
⇒ (î + ĵ + k̂) . (xî + yĵ + zk̂) = 3
⇒ x + y + z = 3 (1)
⇒ x + 2y = 3 [∵ y = z] …(ii)

On subtracting Eq. (i) from Eq. (ii), we get
3y = 2
⇒ y = 23 = z [∵ y = z]

From Eq. (i),
x = 1 + y + 1 = 1 + 23=53
Hence, c⃗ =53i^+23j^+23k^

Question 74.
If a⃗  = î – ĵ + 7k̂ and b⃗  = 5î – ĵ + λk̂, then find the value of λ, so that  and  are perpendicular vectors. (All India 2013)
Answer:
Use the result that if a⃗  and b⃗  are perpendicular, then their dot product should be zero and simplify it.
Given, a⃗  = î – ĵ + 7k̂ and b⃗  = 5î – ĵ + λk̂
Then, a⃗ +b⃗  = (î – ĵ + 7k̂) + (5î – ĵ + λk̂)
= 6î – 2ĵ + (7 + λ) k̂
and a – = (î – ĵ + 7k̂) – (5î – j ̂+ λk̂)
= -4î + (7 – λ)k̂

Since, (a⃗ +b⃗ ) and (a⃗ b⃗ ) are perpendicular
vectors, then (a⃗ +b⃗ )(a⃗ b⃗ ) = 0
⇒ [6î – 2ĵ + (7 + λ)k̂]- [-4î + (7 – λ)k̂] = 0 (1)
⇒ -24 + (7+ X)(7 – X) =0
⇒ 49 – λ2 = 24
⇒ λ2 = 25
∴ λ = ± 5

Question 75.
If p = 5î + λĵ – 3k̂ and q = î + 3ĵ – 5k̂, then find the value of λ, so that p⃗ +q⃗  and p⃗ q⃗  are perpendicular vectors. (All India 2013)
Answer:
λ = ± 1

Question 76.
If a⃗ ,b⃗  and c⃗  are three vectors, such that |a⃗ | = 5, |b⃗ | = 12, |c⃗ | = 13 and a⃗ +b⃗ +c⃗  = 0, then find the value of a⃗ b⃗ +b⃗ c⃗ +c⃗ a⃗ . (Delhi 2012)
Answer:
-169

Question 77.
Let a⃗  = î + 4ĵ + 2k̂, b⃗  = 3î – 2ĵ + 7k̂ and c⃗  = 2î – ĵ + 4k̂. Find a vector p⃗ , which is perpendicular to both a⃗  and b⃗  and p⃗ .c⃗  = 18. (All India 2012,2010)
Answer:
Given vectors are a⃗  = î + 4ĵ + 2k̂,
b⃗  = 3î – 2ĵ + 7k̂
and c⃗  = 2î – ĵ + 4k̂

Let p⃗  = xî + yĵ + zk̂
We have, p⃗  is perpendicular to both a⃗  and b⃗ .
p⃗ a⃗  = 0
⇒ (xî + yĵ + zk̂) – (î + 4ĵ + 7k̂) = 0
⇒ x + 4y + 2z = 0 ………….(i)

and p⃗ b⃗  = 0
⇒ (xî + yĵ + zk̂) . (3î – 2ĵ + 7k̂) = 0
⇒ 3x – 2y + 7z = 0 …(ii)

Also, given ~p-~c =18 (1)
⇒ (xî + yĵ + zk̂) . (2î – ĵ + 4k̂) = 0
⇒ 2x – y + 4z = 18 …(iii)

On multiplying Eq. (i) by 3 and subtracting it from Eq. (ii), we get
– 14y + z = 0 ..(iv)

Now, multiplying Eq. (i) by 2 and subtracting it from Eq. (iii), we get
– 9y = 18
⇒ y = -2

On putting y = -2 in Eq. (iv), we get
-14 (-2) + z = 0
⇒ 28 + z = 0
⇒ z = -28

On putting y = -2 and z = -28 in Eq. (i), we get
x + 4 (-2) + 2 (-28) = 0
⇒ x – 8 – 56 = 0
⇒ x = 64

Hence, the required vector is
p⃗  = xî + yĵ + zk̂
i.e. p⃗  = 64î – 2ĵ – 28k̂

Question 78.
Find a unit vector perpendicular to each of the vectors a⃗ +b⃗  and a⃗ b⃗ , where a = 3î + 2 ĵ + 2k̂ and b = î + 2ĵ – 2k̂. (Delhi 2011)
Answer:
23i^23j^13k^

Question 79.
If a and 6 are two vectors, such that |a⃗ | = 2, |b⃗ | = 1 and a⃗ .b⃗  = 1, then find (3a⃗ 5b⃗ )(2a⃗ +7b⃗ ). (Delhi 2011)
Answer:
Vector Algebra Class 12 Maths Important Questions Chapter 10 83

Question 80.
If vectors a⃗  = 2î + 2ĵ + 3k̂, b⃗  = -î + 2ĵ + k̂ and c⃗  = 3î + ĵ are such that a⃗ +λb⃗  is perpendicular to c⃗ , then find the value λ. (Foreign 2011; All India 2009C)
Answer:
Given, a⃗  = 2î + 2ĵ + 3k̂,
b⃗  = -î + 2ĵ + k̂
and c⃗  = 3î + ĵ

Also, a⃗  + λb⃗  is perpendicular to c⃗ .
∴ (a⃗  + λb⃗ ).c⃗  = 0 …(i) [∵ when [latex][/latex], then a⃗ b⃗  = 0]
Now, a⃗  + λb⃗  = (2î + 2ĵ + 3k̂) + λ (-î + 2ĵ + k̂)
⇒ a⃗  + λb⃗  = î(2 – λ) + ĵ(2 + 2λ) + k̂(3 + λ)
Then, from Bq. (i), we get
[î (2 – λ) + ĵ (2 + 2λ) + k̂(3 + λ)].[3î + ĵ] = 0
⇒ 3(2 – λ) + 1(2+ 2k) = 0
⇒ 8 – λ = 0
∴ λ = 8

Question 81.
Using vectors, find the area of triangle with vertices A (1, 1, 2), 5(2, 3, 5) and C (1, 5, 5). (All India 2011)
Answer:
1261 sq.units

Question 82.
Using vectors, find the area of triangle with vertices A (2, 3, 5), B (3, 5, 8) and C(2, 7, 8). (Delhi 2010C)
Answer:
1261 sq.units

Post a Comment

0 Comments