Cambridge International AS and A Level Mathematics Pure Mathematics 1

Vectors

266

P1^

8

(i) Name the points with the following co-ordinates.

(a) (6, 5, 4)    (b)    (0, 5, 0)    (c)    (6, 2.5, 0)

(d) (0, 2.5, 4)  (e)    (3, 5, 4)

(ii) Use the letters in the diagram to give displacements which are equal to the

following vectors. Give all possible answers; some of them have more than one.

(a)

6

5

4

















(b)

6

0

4

















(c)

0

5

4

















(d)

−

−

















6

5

4

(e)

−















3

25

4

.

ExamPlE 8.9 Figure 8.18 shows a hexagonal prism.

The hexagonal cross-section is regular and consequently A

→

D = 2B

→

C.

A

→

B = p, B

→

C = q and B

→

G = r. Express the following in terms of p, q and r.

(i) A

→

C    (ii)   A

→

D    (iii)  H

→

I    (iv)   I

→

J   

(v) E

→

F    (vi)   B

→

E    (vii)  A

→

H    (viii) F

→

I

SOlUTION

(i) A

→

C   = A

→

B + B

→

C

= p + q

(ii)    A

→

D   = 2B

→

C =     2 q

(iii)   H

→

I = C

→

D

        Since A

→

C + C

→

D   = A

→

D

p + q + C

→

D   =   2 q

        C

→

D   = q − p

So  H

→

I   = q − p

A

p

q

r

D

C

B

G H

I

J

F E

Figure 8.18

A

B q C

p

p + q

p + q

2 q

A

A D

C

B q C

p

p + q

p + q

2 q