# 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),

=> d

_{1}+ 4d_{2}+ 2d_{3}= 0From eq(2),

=> 3d

_{1}– 2d_{2}+ 7d_{3}= 0From eq(3),

=> 2d

_{1}– d_{2}+ 4d_{3}= 15On solving the 3 equations we get,

d

_{1 }= 160/3, d_{2 }= -5/3, and d_{3 }= -70/3,=>

### Q**uestion 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 ,