Add vectors $\to A, \to B$ and $\to C$ each having magnitude of $100$ unit and inclined to the X-axis at angles $45^{\circ}$ , $135^{\circ}$ and $315^{\circ}$ respectively.

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Solution

$y$ component of $\to a = 100 sin 45^{\circ} = \frac{100}{\sqrt{2}}$

$y$ component of $\to b = 100 sin 135^{\circ} = \frac{100}{\sqrt{2}}$

$y$ component of $\to c = 100 sin 315^{\circ} = \frac{- 100}{\sqrt{2}}$

Resultant of $y$ component $= \frac{100}{\sqrt{2}} + \frac{100}{\sqrt{2}} - \frac{100}{\sqrt{2}} = \frac{100}{\sqrt{2}}$ units

$x$ component of $\to a = 100 cos 45^{\circ} = \frac{100}{\sqrt{2}}$

$x$ component of $\to b = 100 cos 45^{\circ} = \frac{- 100}{\sqrt{2}}$

$x$ component of $\to c = 100 cos 45^{\circ} = \frac{100}{\sqrt{2}}$

Resultant of $x$ component $= \frac{100}{\sqrt{2}} - \frac{100}{\sqrt{2}} + \frac{100}{\sqrt{2}} = \frac{100}{\sqrt{2}}$ units

Total resultant of $x$ and $y$ component $= {(\frac{100}{\sqrt{2}})}^{2} + {(\frac{100}{\sqrt{2}})}^{2} = 100$

Now, $tan D = y -$ component $/ x -$ component $= 1$

$D = {tan}^{- 1} (1) = 45^{\circ}$

So, the resultant is $100$ unit and $45^{\circ}$ with $x -$ axis

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