Class 12 RD Sharma Solutions – Chapter 25 Vector or Cross Product – Exercise 25.1 | Set 3

Question 25. If, and , find 

Solution:

We know that,

=> 

=> 

=> 

=> 

=> 

As ,

=> 

=> 

=>

=> 

Thus,

=> 

=>

=> 

Question 26. Find the area of the triangle formed by O, A, B when, 

Solution:

The area of a triangle whose adjacent sides are given by  and  is

=> 

=>

=> 

=> 

=> Area = 

=> Area = 

=> Area = 

=> Area = 

=> Area =  square units.

Question 27. Let ,  and. Find a vector which is perpendicular to bothand and 

Solution:

Given that  is perpendicular to both  and .

=>  ……….(1)

=>  ……….(2)

Also,

=>  …….(3)

Let 

From eq(1),

=> d₁ + 4d₂ + 2d₃ = 0 

From eq(2),

=> 3d₁ – 2d₂ + 7d₃ = 0

From eq(3),

=> 2d₁ – d₂ + 4d₃ = 15 

On solving the 3 equations we get,

d₁ = 160/3, d₂ = -5/3, and d₃ = -70/3, 

=> 

Question 28. Find a unit vector perpendicular to each of the vectors and , where  and .

Solution:

Given that,  and 

Let 

=> 

=> 

=> 

Let 

=> 

=> 

=>

A vector perpendicular to both  and  is,

=> 

=> 

=> 

=> 

To find the unit vector,

=> 

=> 

=> 

=> 

=> 

Question 29. Using vectors, find the area of the triangle with the vertices A(2, 3, 5), B(3, 5, 8), and C(2, 7, 8).

Solution:

Given, A(2, 3, 5), B(3, 5, 8), and C(2, 7, 8)

Let,

=>

=> 

=>

Then,

=> 

=> 

=> 

=> 

=> 

=> 

=> 

=> 

The area of a triangle whose adjacent sides are given by  and  is 

=> 

=> 

=> 

=> Area = 

=> Area = 

=> Area = √61/2

Question 30. If , , are three vectors, find the area of the parallelogram having diagonals  and .

Solution:

Given, , , 

Let,

=> 

=>

=> 

=> 

=>

=> 

=> 

The area of the parallelogram having diagonals  and  is 

=> 

=> 

=>

=> Area = 

=> Area = 

=> Area = 

=> Area = √21/2

Question 31. The two adjacent sides of a parallelogram are and . Find the unit vector parallel to one of its diagonals. Also, find its area.

Solution:

Given a parallelogram ABCD and its 2 sides AB and BC.

By triangle law of addition,

=> 

=> 

=> 

=> 

Unit vector is,

=> 

=> 

=> 

=> 

Area of a parallelogram whose adjacent sides are given is 

=> 

=> 

=> 

Thus area is,

=> Area = 

=> Area = 

=> Area = 

=> Area = 11 √5 square units

Question 32. If either or , then . Is the converse true? Justify with example.

Solution:

Let us take two parallel non-zero vectors  and 

=> 

For example,

and 

=> 

=>

But,

=> 

=> 

Hence the converse may not be true.

Question 33. If , and, then verify that .

Solution:

Given, , and 

=>

=> 

=> 

=>  …..eq(1)

Now,

=> 

=> 

And,

=> 

=>

Thus, 

=> 

=>  …eq(2)

Thus eq(1) = eq(2)

Hence proved.

Question 34(i). Using vectors find the area of the triangle with the vertices A(1, 1, 2), B(2, 3, 5), and C(1, 5, 5).

Solution:

Given, A(1, 1, 2), B(2, 3, 5), and C(1, 5, 5)

=> 

=>

=>

Now 2 sides of the triangle are given by,

=> 

=> 

=> 

=> 

=> 

=> 

=> 

=> 

Area of the triangle whose adjacent sides are given is 

=> 

=> 

=> 

Thus area of the triangle is,

=> Area = 

=> Area = 

=> Area = √61/2

Question 34(ii). Using vectors find the area of the triangle with the vertices A(1, 2, 3), B(2, -1, 4), and C(4, 5, -1).

Solution:

Given, A(1, 2, 3), B(2, -1, 4), and C(4, 5, -1)

=> 

=> 

=> 

Now 2 sides of the triangle are given by,

=>

=>

=> 

=> 

=>

=> 

=> 

=> 

Area of the triangle whose adjacent sides are given is 

=> 

=> 

=> 

Thus area of the triangle is,

=> Area = 

=> Area = 

=> Area = √274/2

Question 35. Find all the vectors of magnitude  that are perpendicular to the plane of and .

Solution:

Given, and 

A vector perpendicular to both  and  is,

=> 

=> 

=> 

Unit vector is,

=> 

=>

=> 

=> 

Now vectors of magnitude  are given by,

=> 

=> Required vectors, 

Question 36. The adjacent sides of a parallelogram are and . Find the 2 unit vectors parallel to its diagonals. Also, find its area of the parallelogram.

Solution:

Given,  and 

=> 

=>

=> 

Unit vector is,

=> 

=> 

=> 

Area is given by ,