First slide

Vectors

Question

$If unit vectors \hat{A} and \hat{B} are inclined at angle θ, then | \hat{A} - \hat{B} | is$

Moderate

A

$2sin \frac{θ}{2}$
B

$2 \cos \frac{θ}{2}$
C

$2 \tan \frac{θ}{2}$
D

$tanθ$

Solution

$| \hat{A} - \hat{B} |^{2} = (\hat{A} - \hat{B}) . (\hat{A} - \hat{B})$
= $\hat{A} . \hat{A} - \hat{A} . \hat{B} - \hat{B} . \hat{A} + \hat{B} . \hat{B}$
= $1 - \hat{A} . \hat{B} - \hat{A} . \hat{B} + 1$
= $2 - 2 cosθ = 2 (1 - cosθ)$
= $2 (2 \sin^{2} \frac{θ}{2}) = 4 \sin^{2} \frac{θ}{2}$
$|\hat{A} - \hat{B}| = 2 \sin \frac{θ}{2}$

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