Cambridge International AS and A Level Mathematics Pure Mathematics 1

Exercise

(^) 8C
275
P1^
8
Substituting gives
cosθ=
×

36

21 78

⇒ θ = 27.2°

!   You must be careful to find the correct angle. To find ∠QPR (see figure 8.23),

you need the scalar product P

→

Q. P

→

R, as in the example above. If you take

Q

→

P. P

→

R, you will obtain ∠Q ́PR, which is (180° − ∠QPR).

ExERCISE 8C  1  Find the angles between these vectors.

(i) 2i +    3 j and     4 i + j (ii)    2 i − j and i +     2 j

(iii) –

• 
  

–

–

1

1

1

2















and  (iv)     4 i + j and i + j

(v) (^)  32  and – 46  (vi)

3

1

6

– 2

 –















and

2  The points A, B and C have co-ordinates (3, 2), (6, 3) and (5, 6), respectively.

(i) Write down the vectors A

→

B and B

→

C.

(ii) Show that the angle ABC is 90°.

(iii) Show that | A

→

B | = | B

→

C |.

(iv) The figure ABCD is a square.

Find the co-ordinates of the point D.

θ^1

4

2 ))

2

5

7 ))

(1, 0, –1)

(2, 4, 1)

(3, 5, 6)

P

Q

R

Figure 8.22

R

Q

P

Q′

θ

Figure 8.23