Chapter 23 Scalar Or Dot Product Of Vectors
Class 12th Maths R S Aggarwal Solution
Exercise 23
Question 1.Find 
when
i. 
and 
ii. 
and
iii. 
and
Answer:
i)
Ans:
ii)
⇒ 
⇒ 
Ans: ⇒ 
iii)
⇒ 
⇒ 
Ans: ⇒ 
Question 2.
Find the value of λ for which 
and 
are perpendicular, where
i. 
and 
ii. 
and
iii. 
and
iv. 
and
Answer:
i)
Since these two vectors are perpendicular,their dot product is zero.
Ans: 
ii)
Since these two vectors are perpendicular,their dot product is zero.
Ans: λ = 3
iii)
Since these two vectors are perpendicular,their dot product is zero.
Ans: λ = - 2
iv)
Since these two vectors are perpendicular,their dot product is zero.
Ans: λ = - 2
Question 3.
i. If 
and 
show that 
is perpendicular to 
ii. If 
and 
then show that and 
are orthogonal.
Answer:
i)
Now 
= (4 × - 2) + (1 × 3) + ( - 1 × - 5) = - 8 + 3 + 5 = 0
Since the dot product of these two vectors is 0,the vector is perpendicular to
Hence,proved.
ii)
Now 
= (6 × 4) + (2 × - 4) + ( - 8 × 2) = 24 - 8 - 16 = 0
Since the dot product of these two vectors is 0,the vector is perpendicular to
Hence,proved that and are orthogonal.
Question 4.
If 
and 
then find the value of λ so that and are orthogonal vectors.
Answer:
(
Now 
Since these two vectors are orthogonal,their dot product is zero.
⇒ (6 × - 4) + ( - 2 × 0) + ( (7 + λ) × (7 - λ)) = 0
⇒ - 24 + 0 + (49 - λ2) = 0
⇒ λ2 = 25
⇒ λ = ±5
Ans: λ = ±5
Question 5.
Show that the vectors
and
are mutually perpendicular unit vectors.
Answer:
Let,
|
| = |
| = |
| = 1
We have to show that :
L.H.S.
= R.H.S.
Hence,showed that vectors are mutually perpendicular unit vectors.
Question 6.
Let 
and 
Find a vector 
which is perpendicular to both and 
and is such that 
Answer:
Let 
the vector which is perpendicular to both and
⇒ 
⇒ 
Solving equations 1,2,3 simultaneously we get
p = 7,q = - 7,r = - 7
∴ 
Ans: 
Question 7.
Let 
and 
Find the projection of (i) on and (ii) on 
Answer:
|| = 
|| = 
Projection of 
Projection of 
Ans: i)
ii)
Question 8.
Find the projection of 
in the direction of 
Answer:
Let,
|
| = 
∴ The projection of 
is:
Ans:10/3
Question 9.
Write the projection of vector 
along the vector 
Answer:
Let,
|| = 
∴ The projection of 
is:
Ans:1
Question 10.
i. Find the projection of on if 
and 
ii. Write the projection of the vector 
on the vector 
Answer:
i)
|| = 
Projection of on
= 
= 
ANS:8/7
ii) Sol:
Let,
|| = 
∴ The projection of 
is:
Ans: 0
Question 11.
Find the angle between the vectors and when
i. 
and 
ii. 
and 
iii. 
and 
Answer:
i) and
|| = 
|| = 
We know that ,
⇒ 
⇒ (3 + 4 + 3) = 14cosθ
⇒ cosθ = 10/14
⇒ cosθ = 5/7
⇒ θ = cos - 1(5/7)
Ans: θ = cos - 1(5/7)
ii) and
|| = 
|| = 
We know that ,
⇒ 
⇒ (6 - 2 + 8) = √336 cosθ
⇒ cosθ = 12/√336
⇒ cosθ = √(144/336)
⇒ θ = cos - 1√ (3/7)
Ans: θ = cos - 1√ (3/7)
iii. and
Ans:
|| =
|| = 
We know that ,
⇒ 
⇒ ( - 1) = 2 cosθ
⇒ cosθ = - 1/2
⇒ θ = cos - 1 - 1/2
⇒ θ = 120⁰
Ans: θ = 120⁰
Question 12.
If 
and 
then calculate the angle between 
and 
Answer:
|
| = 
|
| = 
We know that ,
⇒ 
⇒ (35 - 4) = 50 cosθ
⇒ cosθ = 31/50
⇒ θ = cos - 1(31/50)
Ans: θ = cos - 1(31/50)
Question 13.
If is a unit vector such that 
find 
Answer:
If is a unit vector
⇒ || = 1
⇒ 
⇒ |
|2 - ||2 = 8
⇒ ||2 = 8 + 1 = 9
⇒ || = 3
Ans: || = 3
Question 14.
Find the angles which the vector 
makes with the coordinate axes.
Answer:
If we have a vector = a
+ b
+ c
then the angle with the x - axis = 
the angle with the y - axis = 
the angle with the z - axis = 
Here, = 3 - 6 + 2
then the angle with the x - axis = 
the angle with the y - axis = 
the angle with the z - axis = 
Ans:
Question 15.
Show that the vector 
is equally inclined to the coordinate axes.
Answer:
If we have a vector = a + b + c
then the angle with the x - axis =
the angle with the y - axis =
the angle with the z - axis =
Here, 
then the angle with the x - axis = 
the angle with the y - axis = 
the angle with the z - axis = 
Now since,
∴ the vector is equally inclined to the coordinate axes.
Hence,proved.
Question 16.
Find a vector 
of magnitude 
making an angle π/4 with x - axis, π/2 with y - axis and an acute angle θ with z - axis.
Answer:
|| = 5√2
l = cos α = cos π/4 = 1/√2
m = cos β = cos π/2 = 0
n = cosθ
we know that
l2 + m2 + n2 = 1
since the vector makes an acute angle with the z axis
∴ 
∴ = ||(l + m + n)
∴ = 5√2(1/√2 + 1/√2)
∴ = 5(
)
Ans:
= 5()
Question 17.
Find the angle between 
and 
if 
and 
Answer:
|
| = 
|
| = 
We know that ,
⇒ 
⇒ ( - 5 + 5) = 
cosθ
⇒ cosθ = 0
⇒ θ = cos - 1(0) = π/2
Ans: θ = π/2
Question 18.
Express the vector 
as sum of two vectors such that one is parallel to the vector 
and the other is perpendicular to 
Answer:
∴ 
& 
⇒ 
⇒
⇒
⇒ λ = - 1
∴
⇒ 
⇒ 
⇒
⇒ 
Ans: 
Question 19.
Prove that 
where 
and 
Answer:
⇒ 
⇒ 
Which is not possible hence
Question 20.
If 
and 
find the angle between and
Answer:
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Ans:
Question 21.
Find the angle between and 
when
i. 
and 
ii. 
and 
Answer:
i)
We know that ,
⇒ 
⇒ 
⇒ cosθ = √3/2
⇒ θ = cos - 1(√3/2) = 30⁰ = 
Ans: θ = cos - 1(√3/2) = 30⁰ =
ii)
We know that ,
⇒ 
⇒ 
⇒ cosθ = - 1/2
⇒ θ = cos - 1( - 1/2) = 120⁰ = 
Ans: θ = cos - 1( - 1/2) = 120⁰ =
Question 22.
If 
and 
find 
Answer:
We know that ,
⇒ 
⇒ 
⇒ cosθ = 4/6
⇒ cosθ = 2/3
⇒
⇒ 
⇒ 
⇒ 
Ans: √5
Question 23.
If 
and 
find 
and 
Answer:
().(
) = 8
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Ans:
Question 24.
If 
and 
are unit vectors inclined at an angle θ then prove that:
i. 
ii. 
Answer:
R.H.S:
)
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
= L.H.S
Hence, proved
ii)
R.H.S. = 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ tanθ/2 = L.H.S
Hence,proved.
Question 25.
The dot products of a vector with the vector 
and 
are 0, 5 and 8 respectively. Find the vector.
Answer:
Let the unknown vector be:
∴ 
⇒ a + b - 3c = 0 …(1)
⇒ a + 3b - 2c = 5 …(2)
⇒ 2a + b + 4c = 8 …(3)
Solving equations 1 ,2,3, simultaneously we get:
a = 1,b = 2,c = 1
∴ 
Ans:
Question 26.
If 
and the coordinates of A are (0, - 2, - 1), find the coordinates of B.
Answer:
⇒ 
⇒ 
⇒ 
∴ B(3, - 3,1)
Ans: B(3, - 3,1)
Question 27.
If A(2, 3, 4), B(5, 4, - 1), C(3, 6, 2) and D(1, 2, 0) be four points, show that 
is perpendicular to 
Answer:
Hence,
Question 28.
Find the value of λ for which the vectors 
and 
are perpendicular to each other.
Answer:
Since these two vectors are perpendicular, their dot product is zero.
Ans: 
Question 29.
Show that the vectors 
and 
form a right - angled triangle.
Answer:
⇒ 
Hence, the triangle is a right angled triangle at c
Question 30.
Three vertices of a triangle are A(0, - 1, - 2), B(3, 1, 4) and C(5, 7, 1). Show that it is a right - angled triangle. Also, find its other two angles.
Answer:
∴ 
∴ 
∴
Ans:45°,90°,45°
Question 31.
If the position vectors of the vertices
A, B and C of a ∆ABC be (1, 2, 3), ( - 1, 0, 0) and (0, 1, 2) respectively then find ∠ABC.
Answer:
∴ 
Ans:
= ∠ABC
Question 32.
If and 
are two unit vectors such that 
find 
Answer:
⇒ 
⇒ cosθ = 1/2
∴ 
⇒ 
⇒ 
Ans:
Question 33.
If and are two vectors such that 
then prove that vector 
is perpendicular to the vector
Answer:
⇒ 
⇒ 
⇒ 
NOW,
⇒ 
⇒ 
Hence,
Question 34.
If 
and 
then express in the form 
where 
and 
Answer:
Let b1 = c and b2 = d
∴ 
& 
⇒ 
⇒
⇒
⇒ λ = 5/10 = 1/2
∴
⇒ 
⇒ 
⇒ 
b1 + b2
⇒ 
Ans: