Analysis and Approaches for IBDP Maths Ebook 2 | Page 99

PQR is a triangle such that QR �
� p , RP
� � q and PQ � � r .
(c)
(d)
sin RPQ ˆ sin PQR ˆ sin QRP ˆ
Use (a) to show that � � .
p q r
Show that
p � q � 2 p q cos QRP ˆ � r
2 2 2
[3]
[5]
3. p , q and r are three non-zero vectors.
(a)
Show that
2 2 2 2
r � � p � 2�p�q � q if r ��p �q , � � .
(b) Show that ( p �q �r) �( p�( q �r )) � 0 if ( p�q) �r � q�( r�p ) .
�pq
� �
The vector projection v of p on q is defined by v� q �
2
. �
� q �
(c) Show that v�( p � v)
� 0.
Let � be the acute angle between p and v .
[3]
[4]
[3]
8
(d)
Show that
tan� �
p
2 2
� v
v
2
.
[5]
4. PQR is a triangle such that PQ � � r and PR � � q . N is a point on QR such that
PN � QR . Let QN : NR � � :1, where 0��
� 1.
(a) Express PN � in terms of � , q and r .
[4]
r �( r �q)
(b) Show that � �
q � ( q � r )
. [4]
(c)
Hence, show that
( r �( r �q)) q � ( q�( q �r)) r � r �q PN � 0 .
2
�
[7]
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