Vectors
270
P1^
8
5 In the diagram, ABC is a triangle.
L, M and N are the mid-points of
the sides BC, CA and AB.
A
→
B = p and A
→
C = q
(i) Find, in terms of p and q, B
→
C,
M
→
N, L
→
M and L
→
N.
(ii) Explain how your results from part (i) show you that the sides of triangle
LMN are parallel to those of triangle ABC, and half their lengths.
6 Find unit vectors in the same directions as the following vectors.
(i) 2
3
(ii) 3 i + 4 j (iii) –
2
2
^ (iv)^5 i^ −^12 j
7 Find unit vectors in the same direction as the following vectors.
(i)
1
2
3
(ii) 2 i – 2 j + k (iii) 3 i – 4 k
(iv)
−
−
2
4
3
(v) 5 i – 3 j + 2 k (vi)
4
0
0
8 Relative to an origin O, the position vectors of the points A, B and C are
given by
O
→
A =
2
1
3
, O
→
B =
−
2
4
(^3)
and O
→
C =
−
1
2
1
.
Find the perimeter of triangle ABC.
9 Relative to an origin O, the position vectors of the points P and Q are given
by O
→
P = 3 i + j + 4 k and O
→
Q = i + xj − 2 k.
Find the values of x for which the magnitude of PQ is 7.
10 Relative to an origin O, the position vectors of the points A and B are given by
O
→
A =
4
1
− 2
and O
→
B =
3
2
– 4
.
(i) Given that C is the point such that A
→
C = 2A
→
B, find the unit vector in the
direction of O
→
C.
The position vector of the point D is given by O
→
D =
1
4
k
, where k is a
constant, and it is given that O
→
D = mO
→
A + nO
→
B, where m and n are constants.
(ii) Find the values of m, n and k.
[Cambridge AS & A Level Mathematics 9709, Paper 1 Q9 June 2007]
N
B C
A
M
L