Rajasthan Board RBSE Class 12 Maths Chapter 13 Vector Ex 13.2

Question 1.
If magnitude of two vectors be 4 and 5 units, then find scalar product of them, where angle between them be :
(i) 60°
(ii) 90°
(iii) 30°
Solution:

Question 2.
Find . , if and are as follow :


Solution:

Question 3.
Prove that:

Solution:

Question 4.
If coordinates of P and Q are (3, 4) and (12, 4) respectively, then find ∠POQ where O is origin.
Solution:

Question 5.
For which value of λ, vectors and are mutually perpendicular :

Solution:

Question 6.
Find the projection of the vector

on the vector

Solution:

Question 7.
If

and

then find a vector , so that , , represents the sides of a right angled triangle.
Solution:
Given that

Question 8.
If | + | = | – |, then prove that and are mutually perpendicular vectors.
Solution:
According to question,

Question 9.
If coordinates of points A, B, C and D are (3, 2, 4), (4, 5, -1), (6, 3, 2) and (2, 1, 0) respectively, then prove that lines and are mutually erpendicular.
Solution:
Given that coordinates of points A, B, C and D are (3,2,4), (4,5,-1), (6,3,2) and (2,1,0) respectively.
Then position vectors of A, B, C and D with respect to origin are

Question 10.
For any vector , prove that

Solution:

Question 11.
Use vectors to prove that sum of square of diagonal of a parallelogram is equal to the sum of square of their side.
Solution:
Let OACB is a parallelogram. Taking O as origin, the position vectors of A and B are and respectively.

∴ Sum of the squares of diagonals = sum of the squares of sides.
Hence the sum of square of diagonals of a parallelogram is equal to the sum of square of their sides.