Class 12 RD Sharma Solutions – Chapter 25 Vector or Cross Product – Exercise 25.1 | Set 1

Question 1. If and , find 

Solution:

Given, and  .

=> = 

=> = 

=> = 

=> = 

=> = 

Now, 

=> = 

=> = 

=> = √91

Question 2(i). If  and , find the value of 

Solution:

Given,  and 

=> = 

=> = 

=> = 

=> = 

=> = 

Now, 

=> = 

=> = 

=> = 

Question 2(ii). If  and , find the magnitude of 

Solution:

Given,  and 

=> = 

=> = 

=> = 

=> = 

=> = 

Now, 

=> = 

=> = 

=> = √6

Question 3(i). Find a unit vector perpendicular to both the vectors  and 

Solution:

Given  and 

A vector perpendicular to 2 vectors is given by 

=> = 

=> = 

=> = 

=> = 

=> = 

Unit vector is given by 

=>  = 

=> = 

=> = 3

=> Unit vector is,

=> = 

Question 3(ii). Find a unit vector perpendicular to the plane containing the vectors  and  .

Solution:

 Given, and 

A vector perpendicular to 2 vectors is given by 

=> = 

=> = 

=> = 

=> = 

=> = 

Unit vector is given by 

=> = 

=> = 

=> = 

=> Unit vector is,

=> = 

Question 4. Find the magnitude of vector 

Solution:

Given 

=> 

=> = 

=> = 

=> = 

=> = 

Unit vector is,

=> = 

=> = 

=> = √74

Question 5. If  and , then find 

Solution:

Given, and 

=> = 

=> = 

=> = 

=> = 

=> = 

=> = 

=> = 

=> = 

Now, 

=> = 

=> = 

=> = 

Question 6. If and , find 

Solution:

Given,  and 

=> = 

=> = 

=> = 

=> = 

=> = 

=> = 

=> = 

=> = 

=> = 

=> = 

=> = 

=> = 

=> = 

=> = 

Question 7(i). Find a vector of magnitude 49, which is perpendicular to both the vectors  and 

Solution:

Given,  and 

A vector perpendicular to 2 vectors is given by 

=> = 

=>  = 

=>  = 

=>  = 

=> = 

Magnitude of vector is given by,

=> = 

=> = 

=> = 

=> = 

=> Vector is, 

Question 7(ii). Find the vector whose length is 3 and which is perpendicular to the vector and 

Solution:

Given,  and 

A vector perpendicular to 2 vectors is given by  

=> = 

=> = 

=> = 

=> = 

=> = 

Magnitude of vector is given by,

=> = 

=> = 

=> = 

=> = 27

=> Unit vector is,

=> = 

=> = 

Required vector is, 

=> 

Question 8(i). Find the parallelogram determined by the vectors:  and 

Solution:

Given that, and 

=> Area of the parallelogram is 

=> = 

=> = 

=> = 

=> = 

=> = 

Thus the area of parallelogram is,

=> = 

=> = 

=> Area = 6 square units.

Question 8(ii). Find the parallelogram determined by the vectors:  and .

Solution:

Given that, and 

=> Area of the parallelogram is 

=> = 

=> = 

=> = 

=> = 

=> = 

Thus, the area of parallelogram is,

=> = 

=> = 

=> Area = 

Question 8(iii). Find the area of the parallelogram determined by the vectors:  and 

Solution:

Given that, and 

=> Area of the parallelogram is 

=> = 

=> = 

=> = 

=> = 

=> = 

Thus the area of parallelogram is,

=> = 

=> = 

=> Area = 

Question 8(iv). Find the area of the parallelogram determined by the vectors:  and 

Solution:

Given that,  and 

=> Area of the parallelogram is 

=> = 

=> = 

=> = 

=> = 

=> = 

Thus the area of parallelogram is,

=> = 

=> = 

=> Area = 

Question 9(i). Find the area of the parallelogram whose diagonals are: and 

Solution:

Given, and 

=> Area of the parallelogram is 

=> = 

=> = 

=> = 

=> = 

=> = 

Thus the area of parallelogram is,

=> = 

=> = 

=> Area = 15/2 = 7.5 square units

Question 9(ii). Find the area of the parallelogram whose diagonals are:  and 

Solution:

Given, and 

=> Area of the parallelogram is 

=> = 

=> = 

=> = 

=> = 

=> = 

Thus the area of parallelogram is,

=> = 

=> = 

=> Area = 

Question 9(iii). Find the area of the parallelogram whose diagonals are:  and 

Solution:

Given, and 

=> Area of the parallelogram is 

=> = 

=> = 

=> = 

=> = 

=> = 

Thus the area of parallelogram is,

=> = 

=> = 

=> Area = 

Question 9(iv). Find the area of the parallelogram whose diagonals are:  and 

Solution:

Given, and 

=> Area of the parallelogram is 

=> = 

=> = 

=> = 

=> = 

=> = 

Thus the area of parallelogram is,

=> = 

=> = 

=> Area = 

=> Area = 24.5

Question 10. If  ,  and , compute  and  and verify these are not equal.

Solution:

Given , and 

=> = 

=> = 

=> = 

=> = 

=> = 

=>  = 

=>  = 

=> = 

=> = 

=> = 

=> = 

=> = 

=> = 

=> = 

=> = 

=> = 

=> is not equal to 

=> Hence verified.

Question 11. If ,  and , find 

Solution:

We know that,

=> 

=> 

We know that  is 1, as  is a unit vector

=> 

=> 

=> 

Also,

=> 

And 

=> 

=> 

=> 

=> 

=> 

=> 

Question 12. Given , , , , , being a right-handed orthogonal system of unit vectors in space, show that ,  and  is also another system.

Solution:

To show that ,  and  is a right-handed orthogonal system of unit vectors, we need to prove:

(1) 

=> 

=> 

=>

=>

=>

=>

=>

=> 

=> 

=>

=>

=>

=>

=> 

=>

(2) 

=> 

=>

=>

=>

=>

(3) 

=>

=>

=>

=>

=>

(4)

=>

=>

=>

=>

=>

Hence proved.