Chapter

(^8)
P1^
4  (i) (a) i
(b) 2 i
(c) i − j
(d) −i − 2 j
(ii) | A
→
B | = | B
→
C | = 2 ,
| A
→
D | = | C

→

D | = 5

5  (i) −p + q,^12 p − 21 q, −^12 p, −^12 q

(ii) N
→
M = 12 B
→
C, N
→
L = 21 A
→
C,

M
→
L = 12 A
→
B

  6  (i)

2

13

3

13





















(ii) 35 i +^45 j

(iii)

• 
  


• 
  

1

2

1

2





















(iv) (^135) i –^1213 j
  7  (i)
1
14
2
14
3
14


     


     
(ii) 23 i − 23 j + 13 k
(iii) 35 i −^45 k
(iv)

• 
  


• 
  

2

29

4

29

3

29





     





     

(v) 5
38

i − 3

38

j + 2

38

k

(vi)

1

0

0

















  8  11.74

  9  x = 4 or x = − 2

10  (i) (^17)
2
3
− 6








(ii) m = −2, n = 3, k = − 8
●^ (Page^ 271)
The cosine rule
Pythagoras’ theorem
(^) ● (Page 273)
a
a
b
b
1
2
1
2








.  = a 1 b 1 + a 2 b 2

b

b

a

a

1

2

1

2

















.

= b 1 a 1 + b 2 a 2

These are the same because ordinary

multiplication is commutative.

(^) ● (Page 274)
Consider the triangle OAB with angle
AOB = θ, as shown in the diagram.
cosθ=OA××+OAOB –OBAB
22 2
2
OA^2 = a 1 2 + a 2 2 + a 3 2
OB^2 = b 1 2 + b 2 2 + b 3 2
AB^2 = (b 1 − a 1 )^2 + (b 2 − a 2 )^2 + (b 3 − a 3 )^2
⇒
cos
( )
|||
.
||||
θ=
++

2
2
ab 11 ab 22 ab 33
ab
ab
ab
|
Exercise 8C (Page 275)
  1  (i) 42.3°
(ii) 90°
(iii) 18.4°
(iv) 31.0°
(v) 90°
(vi) 180°
  2  (i) ^31 ,− 31 
(ii) B→A. B→C = 0
(iii) | A→B | = | B→C | = 10
(iv) (2, 5)
  3  (i) P→Q = − 4 i + 2 j; R→Q = 4 i + 8 j
(ii) 26.6°
(iii)  3 i + 7 j
(iv) 53.1°
  4  (i) 29.0°
(ii) 76.2°
(iii) 162.0°
  5  (i) O→Q^ = 3 i + 3 j + 6 k,
P→Q^ = − 3 i + j + 6 k
(ii) 53.0°
  6  (i) − 2
(ii) 40°
(iii) A→B = i − 3 j + (p − 2)k;
p = 0.5 or p = 3.5
  7  (i) −6, obtuse
(ii)
2 3 2 3 1 3

• 
  

























  8  (i) 99°

(ii)  71 (2i − 6 j + 3 k)

(iii) p = −7 or p = 5

  9  (ii) q = 5 or q = − 3

10  (i) P→A = − 6 i − 8 j − 6 k,

P→N = 6 i + 2 j − 6 k

(ii) 99.1°

11  (i)  4 i + 4 j + 5 k, 7.55 m

(ii) 43.7° (or 0.763 radians)

12  (i) P→R =  2 i + 2 j + 2 k,

P

→

Q^ = − 2 i + 2 j + 4 k

(ii) 61.9°

(iii) 12.8 units

θ

O

b a

b – a = (b 1 – a 1 )i +

(b 2 – a 2 )j + (b 3 – a 3 )k

B A