### Question 1. If and , find

**Solution:**

Given, and .

=> =

=> =

=> =

=> =

=> =

Now,

=> =

=> =

=> = âˆš91

### 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,

=> =

=> =

=> = âˆš6

### 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,

=> =

=> =

=> = âˆš74

### 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.