First slide

Scalars and Vectors

Question

If, the resultant of two forces (A + B) and (A - B) is $\sqrt{A^{2} + B^{2}}$ , then the angle between these forces is

Moderate

A

$\cos^{- 1} - \frac{(A^{2} - B^{2})}{A^{2} + B^{2}}$
B

$\cos^{- 1} - \frac{(A^{2} + B^{2})}{(A^{2} - B^{2})}$
C

$\cos^{- 1} [- \frac{A^{2} + B^{2}}{2 (A^{2} - B^{2})}]$
D

$\cos^{- 1} [- \frac{2 (A^{2} + B^{2})}{A^{2} - B^{2}}]$

Solution

Here, $P = (A + B) Q = (A - B)$
and $R = \sqrt{A^{2} + B^{2}}$
$\Rightarrow cosθ = \frac{R^{2} - P^{2} - Q^{2}}{2 PQ} = \frac{(A^{2} + B^{2}) - (A + B)^{2} - (A - B)^{2}}{2 (A + B) (A - B)} = - [\frac{A^{2} + B^{2}}{2 (A^{2} - B^{2})}] θ = \cos^{- 1} [- \frac{A^{2} + B^{2}}{2 (A^{2} - B^{2})}]$

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