# 4.4 Expression of a Vector as the Linear Combination of a Few Vectors

4.4 Expression of a Vector as the Linear Combination of a Few Vectors
1. Polygon Law for Vectors
$\stackrel{\to }{PQ}=\stackrel{\to }{PU}+\stackrel{\to }{UT}+\stackrel{\to }{TS}+\stackrel{\to }{SR}+\stackrel{\to }{RQ}$

2. To prove that two vectors are parallel, we must express one of the vectors as a scalar multiple of the other vector.
For example,

3. To prove that points P, Q and R are collinear, prove one of the following.

Example:
Diagram below shows a parallelogram ABCD. Point Q lies on the straight line AB and point S lies on the straight line DC. The straight line AS is extended to the point T such that AS = 2ST.
It is given that AQ : QB = 3 : 1, DS : SC = 3 : 1,
(a) Express, in terms of

(b) Show that the points Q, C and T are collinear.

Solution: