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

P Q = P U + U T + T S + S R + R Q

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, AB =k CD  or  CD =h AB . . 

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

   PQ =k QR  or  QR =h PQ    PR =k PQ  or  PQ =h PR    PR =k QR  or  QR =h PR


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, AQ =6 a ˜  and  AD = b ˜   
(a) Express, in terms of a ˜  and  b ˜ :   
 (i)  AS    (ii)  QC
(b) Show that the points Q, C and T are collinear.

Solution:
(a)(i)  AS = AD + DS  = AD + AQ AQ:QB= 3:1 and  DS:SC= 3:1 AQ = DS  = b ˜ +6 a ˜  =6 a ˜ + b ˜


(a)(ii)  QC = QB + BC   = 1 3 AQ + AD AQ:QB= 3:1 AQ QB = 3 1 QB= 1 3 AQ and for parallelogram,  BC//AD, BC=AD    = 1 3 ( 6 a ˜ )+ b ˜   =2 a ˜ + b ˜


(b)  QT = QA + AT   = QA + 3 2 AS AS=2ST AT=3ST= 3 2 AS   =6 a ˜ + 3 2 ( 6 a ˜ + b ˜ )   =3 a ˜ + 3 2 b ˜   = 3 2 ( 2 a ˜ + b ˜ )   = 3 2 QC Points Q, C and T are collinear.

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