This document contains 6 presentations on vectors: 1) Identifying equivalent and opposite vectors in a diagram 2) Calculating components of vectors 3) Writing vector expressions in terms of other vectors 4) Illustrating vector addition and subtraction on a grid 5) Expressing vectors in terms of other vectors using properties of midpoints 6) Expressing vectors in terms of other vectors and proving collinearity using properties of parallelograms

1. Unit 38
Vectors
Presentation 1 Equal Vectors
Presentation 2 Components
Presentation 3 Vector Expressions
Presentation 4 Addition and Subtraction of Vectors
Presentation 5 Vector Geometry 1
Presentation 6 Vector Geometry 2

2. Unit 38
38.1 Equal Vectors

3. In the following diagram, state
Which pairs of vectors are equivalent and which pairs of vectors
are opposite directions
Equivalent vectors:
a and b b and i f and g
Vectors in opposite directions
c and b e and j?
???
?

4. Unit 38
38.2 Components

5. Does ?
?
?
?
?
Given that , , calculate
?
?
?
?
?
?
No

6. Unit 38
38.3 Vector Expressions

7. Write down each of the following in terms of d and/or e.
(a)
(b)
(c) ?
??
??
Mark clearly on the diagram
(a) the point P such that
(b) the point Q such that
Q
P
?

8. Unit 38
38.4 Addition and Subtraction of
Vectors

9. Given that
On the grid below, illustrate the following vectors
(i) b – c
(ii) c + a
(iii) a + b
(iv) a + c – b
b - c
- c
b
(i)
(ii)
c + a
c
a
(iii)
a + b
b
a
(iv) a + c - b
a + c - b

10. Unit 38
38.5 Vector Geometry 1

11. Express, in terms of u and v,
(a)
(b)
(c) , where M is the midpoint of AC, ?
?
?

12. Express, in terms of u and v,
(d)
(e) , where N is the midpoint of BD,
(f)
?
?
? ?
?

13. What can you deduce about points M and N?
They are coincident.
Hence and

14. Unit 38
38.6 Vector Geometry 2

15. ) Express in terms of x and y,
(i) (ii) (iii)
olution
In the figure above, ABCD is a parallelogram such that
and . The point P is on DB such that
(i)
(ii)
(iii) ?
?
?
?

16. ?
In the figure above, ABCD is a parallelogram such that
and . The point P is on DB such that
(b) Show that
Solution
?

17. In the figure above, ABCD is a parallelogram such that
and . The point P is on DB such that
(c) Given that E is the midpoint of DC, prove that A, P and E are
collinear.
Solution
so
Hence A, P and E are collinear?
?
??
?
from part (b)
from part (a)