Question

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

Solution

## $$y$$ component of $$\vec{a}= 100\sin{{45}^{\circ}} =\dfrac{100}{\sqrt{2}}$$$$y$$ component of $$\vec{b}= 100\sin{{135}^{\circ}} =\dfrac{100}{\sqrt{2}}$$$$y$$ component of $$\vec{c}= 100\sin{{315}^{\circ}} =\dfrac{-100}{\sqrt{2}}$$Resultant of $$y$$ component$$=\dfrac{100}{\sqrt{2}}+\dfrac{100}{\sqrt{2}}-\dfrac{100}{\sqrt{2}}=\dfrac{100}{\sqrt{2}}$$units$$x$$ component of $$\vec{a}= 100\cos{{45}^{\circ}} =\dfrac{100}{\sqrt{2}}$$$$x$$ component of $$\vec{b}= 100\cos{{45}^{\circ}} =\dfrac{-100}{\sqrt{2}}$$$$x$$ component of $$\vec{c}= 100\cos{{45}^{\circ}} =\dfrac{100}{\sqrt{2}}$$Resultant of $$x$$ component$$=\dfrac{100}{\sqrt{2}}-\dfrac{100}{\sqrt{2}}+\dfrac{100}{\sqrt{2}}=\dfrac{100}{\sqrt{2}}$$unitsTotal resultant of $$x$$ and $$y$$ component$$={\left(\dfrac{100}{\sqrt{2}}\right)}^{2}+{\left(\dfrac{100}{\sqrt{2}}\right)}^{2}=100$$Now, $$\tan{D}=y-$$component $$/x-$$ component$$=1$$$$D={\tan}^{-1}{\left(1\right)}={45}^{\circ}$$So, the resultant is $$100$$ unit and $${45}^{\circ}$$ with $$x-$$axisPhysics

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