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Systems of Particles and Rotational Motion

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Question : 4
Total: 33
Show that the area of the triangle contained between the vectors
→
a
 and
→
b
 and is one half of the magnitude of
→
a
×
→
b
Solution:  
Let
→
a
 be represented by O
→
P
 and
→
b
 be represented by O
→
Q
 . Let ∠POQ=θ, figure complete the parallelogram OPRQ.
Join PQ. Draw QN ⊥ OP.
In ΔOQN
sin‌θ=
QN
OQ
=
QN
b

QN=b‌sin‌θ
Now, by definition,
|
→
a
×
→
b
|=ab‌sin‌θ=(OP)(QN)
=
2(OP)(QN)
2
=2× area of ΔOPQ
∴ area of ΔOPQ =
1
2
|
→
a
×
→
b
|, which was to be proved.
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