VectorsP1^
8
ExamPlE 8.10 Relative to an origin O, the position vectors of the points A, B and C are given byO
→
A =−
−
2
3
2
, O
→
B =0
1
− 3
and O→
C =−
2
3
1
.
(i) Find the unit vector in the direction A→
B.
(ii) Find the perimeter of triangle ABC.SOlUTION
For convenience call O→
A = a, O→
B = b
and O→
C = c. (i) A→
B = b − a =0
1
3
2
3
2
2
2
− 1
−
−
−
=−
−
To find the unit vector in the direction A→
B, you need to divide A→
B by its
magnitude.
| A
→
B | =+−+ −
=
=22 1
9
3
(^222) () ()
So the unit vector in the direction A
→
B is (^13)
2 3 2 3 1 3
2
2
1
−
−
=−
−
(ii) The perimeter of the triangle is given by | A→
B | + | A→
C | + | B→
C |.A
→
C = c − a =−
−
−
−
=
2
3
1
2
3
2
0
0
3
⇒ | A
→
C | = 00322 + +^2
= 3B
→
C = c − b =−
−
−
=
−
2
3
1
0
1
3
2
2
4
⇒ | B
→
C | = ()−+ 2222 + 42
= 24
Perimeter of ABC = | A→
B | + | A→
C | + | B→
C |
= 3 + 3 + 24
= 10.9This is the
magnitude of A
→
B.