NCERT Solutions for Chapter 10 Vector Algebra Class 12 Maths Miscellaneous Exercise

Class 12 Maths NCERT Solutions for Chapter 10 Vector Algebra Miscellaneous Exercise

Vector Algebra Miscellaneous Exercise Solutions

1. Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.

Solution

If  r⃗ is a unit vector in the XY-plane, then  r⃗ = cosθi ^ + sinθj ^   .
Here, θ is the angle made by the unit vector with the positive direction of the x-axis.
Therefore, for θ = 30° . 
 r⃗ = cosθi ^ + sinθj ^ = (√3/2)i ^ + (1/2)j ^
Hence, the required unit vector is (√3/2) i ^ + (1/2)j ^.

2. Find the scalar components and magnitude of the vector joining the points P(x₁, y₁, z₁ ) and Q(x₂, y₂, z₂) .

Solution

The vector joining the points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) can be obtained by, 

Hence, the scalar components and the magnitude of the vector joining the given points are respectively {(x₂ - x₁), (y₂ - y₁), (z₂ - z₁)} 
and 

3. A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of north and stops. Determine the girl’s displacement from her initial point of departure.

Solution

Let O and B be the initial and final positions of the girl respectively. 
Then, the girl's position can be shown as : 

Now, we have:

By the triangle law of vector addition, we have :  

Hence, the girl's displacement from her initial point of departure is (-5/2) i ^ + (3√3/2) j ^

4. If a⃗ = b⃗ + c⃗, then is it true that |a⃗| = |b⃗| + |c⃗| ? Justify your answer.

Solution

Now, by the triangle law of vector addition, we have a⃗ = b⃗ + c⃗. 
It is clearly known that | a⃗|, |b⃗|, and |c⃗| represent the sides of ΔABC. 

Also, it is known that the sum of the lengths of any two sides of a triangle is greater than the third side. 
∴ | a⃗| < |b⃗| + |c⃗|
Hence, it is not true that |a⃗| = |b⃗| + |c⃗| .

5. Find the value of x for which x(i ^+ j ^+ k ^) is a unit vector. 

Solution

6. Find a vector of magnitude 5 units, and parallel to the resultant of the vectors a⃗ = 2i ^+3j ^-k^ and b⃗ = i ^- 2j ^+ k^ .

Solution

We have, 

a⃗ = 2i ^ + 3j ^ - k^  and b⃗ = i ^ - 2j ^ + k^ 
Let c⃗ be the resultant of a⃗ and b⃗. 
Then, 

Hence, the vector of magnitude 5 units and parallel to the resultant of vectors    a⃗ and  b⃗  is 

7. If a⃗ = i ^+ j ^+ k^, b⃗ = 2i ^- j ^+3k^ and c⃗ = i ^- 2j ^+k^ find a unit vector parallel to the vector 2a⃗ - b⃗ +3c⃗ .

Solution

We have, 

8. Show that the points A (1, –2, –8), B (5, 0, –2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.

Solution

The given points are A (1, –2, –8), B (5, 0, –2) and C (11, 3, 7) . 

Thus, the given points A, B, and C are collinear. 
Now, let point B divide AC in the ratio λ : 1. Then, we have :  

On equating the corresponding  components, we get :  
5(λ + 1) = 11λ + 1
⇒ 5λ + 5 =11λ + 1
⇒ 6λ = 4
⇒ λ = 4/6  = 2/3
Hence, point B divides AC in the ratio 2 : 3.

9. Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are P(2a⃗ + b⃗) and Q(a⃗ - 3b⃗) externally in the ratio 1 : 2. Also, show that P is the mid point of the line segment RQ. 

Solution

It is given that 
It is given that point R divides a line segment joining two points P and Q externally in the ratio 1 : 2. Then, on using the section formula, we get :  

Therefore, the position vector of point R is 
Position vector of the mid - point of RQ = 

Hence, P is the mid - point of the line segment RQ. 

10 . The two adjacent sides of a parallelogram are 2i ^ - 4j ^ + 5k^, and i ^-2j ^-3k^ . Find the unit vector parallel to its diagonal. Also, find its area. 

Solution

Adjacent sides of a parallelogram are given as: a⃗ = 2i ^ -4j ^+5k^  and b⃗ = i ^-2j ^+3k^  . 
Then, the diagonal of a parallelogram is given by  a⃗ +  b⃗ .

∴ Area of parallelogram ABCD = | a⃗ + b⃗ |

Hence, the area of the parallelogram is 11√5 square units. 

11. Show that the direction cosines of a vector equally inclined to the axes OX, OY, and OZ are 1/√3, 1/√3, 1/√3 .

Solution

Let a vector be equally inclined to axes OX, OY, and OZ at angle α. 
Then, the direction cosines of the vector are cos α, cos α and cos α. 

Now,
cos² α + cos² α + cos² α = 1
⇒ 3cos² α = 1 
⇒ cos α  = 1/√3 
Hence, the direction cosines of the vector which are equally inclined to the axes are 1/√3, 1/√3, 1/√3. 

12. Let a⃗ = i ^+4j ^+2k^ , b⃗=3i ^-2j ^+7k^  and c⃗ = 2i ^- j ^+4k^ . Find a vector d⃗ which is perpendicular to both a⃗ and b⃗, and c⃗ . d⃗ = 15. 

Solution

Let  
Since d⃗  is perpendicular to both a⃗ and b⃗, we have :  

13. The scalar product of the vector i ^+ j ^+ k^, with a unit vector along the sum of vectors 2i ^+4j ^-5k^ and λi ^+2j ^+3k^ is equal to one. Find the value of λ . 

Solution

⇒ λ² + 4 λ + 44 = (λ + 6)² 
⇒  λ² + 4λ + 44 = λ² + 12λ + 36
⇒  8λ = 8 
⇒ λ = 1

Hence, the value of λ is 1. 

14. If  a⃗, b⃗, c⃗ are mutually perpendicular vectors of equal magnitudes, show that the vector a⃗ + b⃗ + c⃗  is equally inclined to a⃗, b⃗ and c⃗ . 

Solution

Since, a⃗ , b⃗ and c⃗  are mutually perpendicular vectors, we have 

Let vector a⃗+b⃗ +c⃗ be inclined to a⃗, b⃗ and c⃗ at angles θ₁, θ₂, θ₃ respectively . 
Then , we have :  

Now, as |a⃗|+|b⃗|+ |c⃗|, cos θ₁ = cos θ₂ = cosθ₃.
∴  θ₁ =  θ₂ = θ₃  
Hence, the vector (a⃗ +b⃗ + c⃗) is equally inclined to a⃗, b⃗, and c⃗. 

15. Prove that (a⃗ +b⃗) . (a⃗ +b⃗) = |a⃗|² +|b⃗|² if and only if a⃗ . b⃗ are perpendicular, given a⃗ ≠ 0⃗ , b⃗ ≠ 0⃗ 

Solution

16. If θ is the angle between two vectors a⃗ and b⃗, then a⃗ . b⃗ ≥ 0 only when  
(A) 0 < θ < π/2
(B) 0 ≤ θ ≤ π/2 
(C)  0 < θ < π
(D)  0 ≤ θ ≤ π

Solution

Let θ be the angle between two vectors  a⃗ and b⃗ .
Then, without loss of generality, a⃗ and b⃗ are non-zero vectors so that |a⃗| and |b⃗| are positive. 

Hence, a⃗ . b⃗  ≥ 0 when 0 ≤ θ ≤ π/2  . 
The correct answer is B.

17. Let  a⃗ and  b⃗  be two unit vectors and θ is the angle between them. Then a⃗ + b⃗ is a unit vector if ____ 
(A) θ = π/4 
(B) θ = π/3
(C) θ = π/2 
(D) θ = 2π/3 

Solution

Let a⃗ and b⃗ be two unit vectors and θ be the angle between them. 

Then, | a⃗| = |b⃗| = 1. 
Now, a⃗ + b⃗ is a unit vector if |a⃗ +b⃗| = 1. 

Hence, a⃗ + b⃗ is a unit vector if θ = 2π/3. 

18.  The value of is 

(A) 0 
(B) -1
(C) 1
(D) 3

Solution

The correct answer is C. 

19. If θ is the angle between any two vectors a⃗ and b⃗, then |a⃗ . b⃗| = |a⃗ × b⃗| when θ is equal to _____. 
(A) 0 
(B) π/4 
(C) π/2 
(D) π 

Solution

Let θ be the angle between two vectors  a⃗ and b⃗ .
Then, without loss of generality, a⃗ and b⃗  are non-zero vectors, so that | a⃗ | and | b⃗|are positive. 

Hence, | a⃗ . b⃗| = | a⃗ × b⃗| when θ is equal to π/4.

The correct answer is B.