## Vectors Exercise

If a = , b = c = , d =  and e =

1. Find the following and leave the answer in matrix form:

 a a + b b b − c c 5c d -3e e e + 2c f 2a + 3c g d − 4e h 3b − 2e i a + b + c j 2(a + b − e)

2. Find the length or magnitude of a, b, c, d and e. Give answers to 3 significant figures where appropriate.

3. Illustrate on a diagram, the following vector additions and subtractions. Draw the resultant vector and label it r.

 a a + c b b + c c a + b + c d a − b e b − e f 3c + 4d g 2d − 3e h 4b − d i a + b − c j a + b + c + d + e

4. Express a, b, c, d, and e as unit vectors.

5. Calculate the following using unit vector notation. A diagram is not needed. Check your answers with those calculated in question 1.

 a a + b b b − c c 5c d -3e e e + 2c f 2a + 3c g d − 4e h 3b − 2e i a + b + c j 2(a + b − e)

6. Find the number p such that p =

7. Find numbers s and t such that

8. a. If p = 6i -2jq = 3j -2i and r = 13j − 4i, show that there is a number s such that p + sq = r.

9. The points C and D have coordinates (4, 2) and (2, 3). Find the coordinates of the points E, F, G, H, I and J given by the following vector equations.

 a e = 4c b f = -2d c g = c − d d h = d - 3c e i = d + 3c f j = 1/2(d + 3c)

10. A is the point (2, 3) and B is the point (3, -2)

a. What are the position vectors of A and B as column vectors?

b. Write and as unit vectors and find BA.

c. If = 2a − 3b, find the coordinates of c. Show on a diagram.

d. If a point D is the midpoint of OC find its coordinates.