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8.4 Expression of a Vector as the Linear Combination of a Few Vectors

8.4 Expression of a Vector as the Linear Combination of a Few Vectors
1. Polygon Law for Vectors

PQ=PU+UT+TS+SR+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, AB=kCD or CD=hAB. . 

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

  PQ=kQR or QR=hPQ  PR=kPQ or PQ=hPR  PR=kQR or QR=hPR


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=6a˜ 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+AQAQ:QB= 3:1 and DS:SC= 3:1AQ=DS =b˜+6a˜ =6a˜+b˜


(a)(ii) QC=QB+BC  =13AQ+ADAQ:QB= 3:1AQQB=31QB=13AQand for parallelogram, BC//AD, BC=AD   =13(6a˜)+b˜  =2a˜+b˜


(b) QT=QA+AT  =QA+32ASAS=2STAT=3ST=32AS  =6a˜+32(6a˜+b˜)  =3a˜+32b˜  =32(2a˜+b˜)  =32QCPoints Q, C and T are collinear.

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