### Question 13. If , and , find

**Solution:**

We know that,

=>

=>

=>

=>

=>

Also,

=>

And

=>

=>

=>

=>

=>

=>

### Question 14. Find the angle between 2 vectors and , if

**Solution:**

Given

=>

=>, as is a unit vector.

=>

=>

=>

### Question 15. If , then show that , where m is any scalar.

**Solution:**

Given that

=>

=>

=>

Using distributive property,

=>

If two vectors are parallel, then their cross-product is 0 vector.

=> and are parallel vectors.

=>

Hence proved.

### Question 16. If, and , find the angle between and

**Solution:**

Given that,, and

We know that,

=>

=>

=>

=>

=>

=>

=>

=>

### Question 17. What inference can you draw if and

**Solution:**

Given, and

=>

=>

Either of the following conditions is true,

1.

2.

3.

4.is parallel to

=>

=>

Either of the following conditions is true,

1.

2.

3.

4. is perpendicular to

Since both these conditions are true, that implies atleast one of the following conditions is true,

1.

2.

3.

### Question 18. If , and are 3 unit vectors such that , and . Show that ,and form an orthogonal right handed triad of unit vectors.

**Solution:**

Given, , and

As,

=>

=> is perpendicular to both and .

Similarly,

=> is perpendicular to both and

=> is perpendicular to both and

=> , and are mutually perpendicular.

As, , and are also unit vectors,

=> , and form an orthogonal right-handed triad of unit vectors

Hence proved.

### Question 19. Find a unit vector perpendicular to the plane ABC, where the coordinates of A, B, and C are A(3, -1, 2), B(1, -1, 3), and C(4, -3, 1).

**Solution:**

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

Let,

=>

=>

=>

Plane ABC has two vectors and

=>

=>

=>

=>

=>

=>

=>

=>

A vector perpendicular to both and is given by,

=>

=>

=>

=>

=>

To find the unit vector,

=>

=>

=>

=>

### Question 20. If a, b and c are the lengths of sides BC, CA and AB of a triangle ABC, prove that and deduce that

**Solution:**

Given that , and

From triangle law of vector addition, we have

=>

=>

=>

=>

=>

=>

=>

=>

=>

=>

=>

=>

=>

=>

=>

Similarly,

=>

=>

=>

=>

=>

=>

=>

=>

=>

=>

=>

Hence proved.

### Question 21. If and , then find . Verify that and are perpendicular to each other.

**Solution:**

Given, and

=>

=>

=>

=>

=>

Two vectors are perpendicular if their dot product is zero.

=>

=>

=>

=>

Hence proved.

### Question 22. If and are unit vectors forming an angle of , find the area of the parallelogram having and as its diagonals.

**Solution:**

Given and forming an angle of .

Area of a parallelogram having diagonals and is

=>

=>

=>

Thus area is,

=> Area =

=> Area =

=> Area =

=> Area =

=> Area =

=> Area =

=> Area =

=> Area =

=> Area = square units

### Question 23. For any two vectors and , prove that

**Solution:**

We know that,

=>

=>

=>

=>

=>

=>

=>

=>

=>

Hence proved.

### Question 24. Define and prove that , where is the angle between and

**Solution:**

Definition of

:Let and be 2 non-zero, non-parallel vectors. Then , is defined as a vector with the magnitude of , and which is perpendicular to both the vectors and .We know that,

=>

=>

=> ……………..(eq.1)

And as,

=>

=>

Substituting in (eq.1),

=>

=>