First slide

Vectors

Question

At what angle must the two forces (x + y) and (x $-$ y) act, so that the resultant may be $\sqrt{x^{2} + y^{2}} ?$

Moderate

A

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

$\cos^{- 1} [- \frac{- 2 (x^{2} - y^{2})}{x^{2} + y^{2}}]$
C

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

$\cos^{- 1} [- \frac{(x^{2} - y^{2})}{(x^{2} + y^{2})}]$

Solution

Let the two forces be A and B.
Resultant, $R^{2} = A^{2} + B^{2} + 2 A B \cos θ$ …(i)
Substituting, $A = (x + y), B = (x - y)$
and $R = \sqrt{x^{2} + y^{2}}$
On substituting these values in Eq. (i), we get
$θ = \cos^{- 1} [- \frac{(x^{2} + y^{2})}{2 (x^{2} - y^{2})}]$

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