NCERT Solutions for 12th Class Maths: Chapter 10-Vector Algebra
NCERT Solutions for 12th Class Maths: Chapter 10-Vector Algebra

Class 12: Maths Chapter 10 solutions. Complete Class 12Maths Chapter 10 Notes.

NCERT Solutions for 12th Class Maths: Chapter 10-Vector Algebra

Class 12:Maths Chapter 10 solutions. Complete Class 12Maths Chapter 10 Notes.

Ex 10.1 Class 12 Maths Question 1.
Represent graphically a displacement of 40km, 30° east of north.
Solution:
A line segment of 2 cm is drawn on the right of OY making an angle of 30° with it. OP = 40 km,
scale 1cm = 20 km. Vector 

 represents displacement of 40 km 30° east of north.

Ex 10.1 Class 12 Maths Question 2.
Classify the following measures as scalars and vectors.
(i) 10 kg
(ii) 2 metres north- west
(iii) 40°
(iv) 40 watt
(v) 10-19 coulomb
(vi) 20 m/sec².
Solution:
(i) Mass-scalar
(ii) Directed distance-vector
(iii) Temperature-scalar
(iv) Rate of electricity-scalar
(v) Electric charge-vector
(vi) Acceleration-vector

Ex 10.1 Class 12 Maths Question 3.
Classify the following as scalar and vector quantities
(i) time period
(ii) distance
(iii) force
(iv) velocity
(v)work.
Solution:
Scalar Quantity: (i) time period (ii) distance (v) work.
Vector Quantity: (iii) force (iv) velocity

Ex 10.1 Class 12 Maths Question 4.
In a square, identify the following vectors
(i) Co-initial
(ii) Equal
(iii) collinear but not equal
Solution:
(i) Co initial vectors are 


(ii) Equal Vectors are 


(iii) Collinear but not equal vectors are 


Ex 10.1 Class 12 Maths Question 5.
Answer the following as true or false:
(i) 

 are collinear.
(ii) Two collinear vectors are always equal in magnitude.
(iii) Two vectors having same magnitude are collinear.
(iv) Two collinear vectors having the same magnitude are equal.
Solution:
(i) True
(ii) False
(iii) False
(iv) False.

Ex 10.2 Class 12 Maths Question 1.
Compute the magnitude of the following vectors:



Solution:



Ex 10.2 Class 12 Maths Question 2.
Write two different vectors having same magnitude.
Solution:

Such possible answers are infinite

Ex 10.2 Class 12 Maths Question 3.
Write two different vectors having same direction.
Solution:
Let the two vectors be



Hence vectors 

 have the same direction but different magnitude

Ex 10.2 Class 12 Maths Question 4.
Find the values of x and y so that the vectors 

 are equal.
Solution:
We are given 


If vectors are equal, then their respective components are equal. Hence x = 2, y = 3.

Ex 10.2 Class 12 Maths Question 5.
Find the scalar and vector components of the vector with initial point (2,1) and terminal point (-5,7).
Solution:
LetA(2, 1) be the initial point and B(-5,7) be the terminal point 


∴The vector components are 

 and scalar components are – 7 and 6.

Ex 10.2 Class 12 Maths Question 6.
Find the sum of three vectors:


Solution:

Ex 10.2 Class 12 Maths Question 7.
Find the unit vector in the direction of the vector

Solution:


Ex 10.2 Class 12 Maths Question 8.
Find the unit vector in the direction of vector 

, where P and Q are the points (1,2,3) and (4,5,6) respectively.
Solution:
The points P and Q are (1, 2, 3) and (4, 5, 6) respectively


Ex 10.2 Class 12 Maths Question 9.
For given vectors 

 find the unit vector in the direction of the vector 


Solution:


Ex 10.2 Class 12 Maths Question 10.
Find a vector in the direction of 

 which has magnitude 8 units.
Solution:
The given vector is 


Ex 10.2 Class 12 Maths Question 11.
Show that the vector 

 are collinear.
Solution:



vector 

 have the same direction they are collinear.

Ex 10.2 Class 12 Maths Question 12.
Find the direction cosines of the vector 

Solution:
let 


Now a = 1,b = 2,c = 3

Ex 10.2 Class 12 Maths Question 13.
Find the direction cosines of the vector joining the points A (1,2, -3) and B(-1, -2,1), directed from A to B.
Solution:
Vector joining the points A and B is


Ex 10.2 Class 12 Maths Question 14.
Show that the vector 

 are equally inclined to the axes OX, OY, OZ.
Solution:
Let 

 , Direction cosines of vector 

 are


which shows that the vector a is equally inclined to the axes OX, OY, OZ.

Ex 10.2 Class 12 Maths Question 15.
Find the position vector of a point R which divides the line joining the points whose positive vector are 

 in the ratio 2:1
(i) internally
(ii) externally.
Solution:
(i) The point R which divides the line joining the point 

 in the ratio m : n

Ex 10.2 Class 12 Maths Question 16.
Find position vector of the mid point of the vector joining the points P (2,3,4) and Q (4,1, -2).
Solution:
Let 


Ex 10.2 Class 12 Maths Question 17.
Show that the points A, B and C with position vector 

 respectively form the vertices of a right angled triangle.
Solution:


Ex 10.2 Class 12 Maths Question 18.
In triangle ABC (fig.), which of the following is not


(a) 


(b) 


(c) 


(d) 

Solution:
We know that

Hence option (c) is not correct

Ex 10.2 Class 12 Maths Question 19.
If 

 are two collinear vectors then which of the following are incorrect:
(a) 

, for some scalar λ.
(b) 

(c) the respective components of 

 are proportional.
(d) both the vectors 

 have same direction, but different magnitudes.
Solution:
Options (d) is incorrect since both the vectors 

 , being collinear, are not necessarily in the same direction. They may have opposite directions. Their magnitudes may be different.

Ex 10.3 Class 12 Maths Question 1.
Find the angle between two vectors 

 with magnitudes √3 and 2 respectively, and such that 


Solution:
Angle θ between two vectors 

,

Ex 10.3 Class 12 Maths Question 2.
Find the angle between the vectors 


Solution:
Let 


Let θ be the angle between 

,

Ex 10.3 Class 12 Maths Question 3.
Find the projection of the vector 

, on the line represented by the vector 

,
Solution:
let 


Ex 10.3 Class 12 Maths Question 4.
Find the projection of the vector 

 on the vector 


Solution:
let 

 then

Ex 10.3 Class 12 Maths Question 5.
Show that each of the given three vectors is a unit vector 

 Also show that they are mutually perpendicular to each other.
Solution:


Ex 10.3 Class 12 Maths Question 6.


Solution:
Given 


Ex 10.3 Class 12 Maths Question 7.
Evaluate the product :


Solution:

Ex 10.3 Class 12 Maths Question 8.
Find the magnitude of two vectors 

 having the same magnitude and such that the angle between them is 60° and their scalar product is 


Solution:
We know that 


Ex 10.3 Class 12 Maths Question 9.
Find 

 , if for a unit vector 


Solution:
Given

Ex 10.3 Class 12 Maths Question 10.
If 

 such that 

 , then find the value of λ.
Solution:
Given

Ex 10.3 Class 12 Maths Question 11.
Show that 

 for any two non-zero vectors 

Solution:

 are any two non zero vectors

Ex 10.3 Class 12 Maths Question 12.
If 

, then what can be concluded about the vector 

 ?
Solution:

=> 

 = 0
Hence b is any vector.

Ex 10.3 Class 12 Maths Question 13.
If 

 are the unit vector such that 

 , then find the value of 


Solution:
We have


Ex 10.3 Class 12 Maths Question 14.
If either vector 

 then 

. But the converse need not be true. Justify your answer with an example.
Solution:
Given: 

To prove: 


Ex 10.3 Class 12 Maths Question 15.
If the vertices A,B,C of a triangle ABC are (1,2,3) (-1,0,0), (0,1,2) respectively, then find ∠ABC.
Solution:
Let O be the origin then.

Ex 10.3 Class 12 Maths Question 16.
Show that the points A (1,2,7), B (2,6,3) and C (3,10, -1) are collinear.
Solution:
The position vectors of points A, B, C are

Ex 10.3 Class 12 Maths Question 17.
Show that the vectors 

 and 

 from the vertices of a right angled triangle.
Solution:
The position vectors of the points A, B and C are

 and 


Ex 10.3 Class 12 Maths Question 18.
If 

 is a non-zero vector of magnitude ‘a’ and λ is a non- zero scalar, then λ 

is unit vector if
(a) λ = 1
(b) λ = – 1
(c) a = |λ|
(d) a = 


Solution:

Given : 

 is a unit vectors

Ex 10.4 Class 12 Maths Question 1.
Find 


Solution:
Given

Ex 10.4 Class 12 Maths Question 2.
Find a unit vector perpendicular to each of the vector 

, where 


Solution:
we have


Ex 10.4 Class 12 Maths Question 3.
If a unit vector 

 makes angle 

 and an acute angle θ with 

,then find θ and hence the components of 

.
Solution:


Ex 10.4 Class 12 Maths Question 4.
Show that 


Solution:
LHS = 


Ex 10.4 Class 12 Maths Question 5.
Find λ and μ if


Solution:

Ex 10.4 Class 12 Maths Question 6.
Given that 

. What can you conclude about the vectors 

 ?
Solution:

Ex 10.4 Class 12 Maths Question 7.
Let the vectors 

 are given 

. Then show that 

Solution:
Given

 are given 


Ex 10.4 Class 12 Maths Question 8.
If either 

.Is the
converse true? Justify your answer with an example.
Solution:


Ex 10.4 Class 12 Maths Question 9.
Find the area of the triangle with vertices A (1,1,2), B (2,3,5) and C (1,5,5).
Solution:
A (1,1,2), B (2,3,5) and C (1,5,5).

Ex 10.4 Class 12 Maths Question 10.
Find the area of the parallelogram whose adjacent sides are determined by the vectors 


Solution:
We have 


Ex 10.4 Class 12 Maths Question 11.
Let the vectors

 such that 

 then 

 is a unit vector if the angle between 

 is
(a) 


(b) 


(c) 


(d) 


Solution:
Given



Ex 10.4 Class 12 Maths Question 12.
Area of a rectangles having vertices



(a) 

sq units
(b) 1sq.units
(c) 2sq.units
(d) 4sq.units
Solution:

NCERT Solutions for 12th Class Maths: Chapter 10: Download PDF

NCERT Solutions for 12th Class Maths: Chapter 10-Vector Algebra

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