Analysis and Approaches for IBDP Maths Ebook 2 | Page 97

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Paper 1 Section B – Vector Algebra
Example
p , q and r are three non-zero vectors.
(a) Show that p and q�r are perpendicular if a( p�q) �b( q�r) � c( r�p)
� 0, where
a , b , c � 0 .
p , q , r and s are the position vectors of the points P , Q , R and S respectively.
[4]
(b) If PQRS is a parallelogram, show that the area of PQRS is p�q � q�s � s�p .
(c) Hence, show that p�q �q�r �r�p � 0 if P , Q and R are collinear.
[5]
[6]
Solution
8
(a) a( p�q) �b( q�r) � c( r�p)
� 0
(b)
a( p�q) �p �b( q�r) �p � c( r�p)
�p � 0�p M1
a( p�( p�q)) �b( p�( q�r)) � c( p�( r�p )) � 0 A1
a( 0) �b( p�( q�r)) � c( 0 ) � 0
A1
b( p�( q�r
)) � 0
p�( q�r ) � 0
A1
Thus, p and
The area of PQRS
q�r are perpendicular.
AG
[4]
� �
� PQ� PS
M1
� ( q �p) �( s �p )
A1
� ( q �p) �s � ( q �p)
�p
� q�s �p�s �q�p � p�p M1
� q�s �p�s �q�p � 0 A1
� q�s � ( �s�p) � ( �p�q )
A1
� p�q � q�s � s�p AG
[5]
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