Add vectors vector A, vector B and vector C each having magnitude of

Vectors $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are oriented at $45^{\circ}, 135^{\circ}$ and $315^{\circ}$ respectively.

$|A|=|B|=|C|=100$ units

Let $A=A_{x} \mathbf{i}+A_{y} \mathbf{j}+A_{z} \mathbf{k}, B=B_{x} \mathbf{i}+B_{y} \mathbf{j}+B_{z} \mathbf{k}$, and $C=C_{x} \mathbf{i}+C_{y} \mathbf{j}+C_{z} \mathbf{k}$, and we can write that, $A_{x}=C_{x}=100 \cos \left(45^{\circ}\right)=100 / \sqrt{2}$, by considering their components

$\mathrm{B}_{\mathrm{x}}=-100 / \sqrt{2}$

Now Ay $=100 \sin \left(45^{\circ}\right)=100 / \sqrt{2}$,

\begin{aligned}

&\text { By }=100 \sin \left(135^{\circ}\right)=100 / \sqrt{2} \\

&\text { Similarly, Cy= }-100 / \sqrt{2} \\

&\text { Net } x \text { component }=100 / \sqrt{2}+100 / \sqrt{2}-100 / \sqrt{2}=100 / \sqrt{2} \\

&\text { Net } y \text { component }=100 / \sqrt{2}+100 / \sqrt{2}-100 / \sqrt{2}=100 / \sqrt{2} \\

&R^{2}=x^{2}+y^{2}=100^{2} \\

&R=100 \text { and } \tan \phi=(100 / \sqrt{2}) /(100 / \sqrt{2})=1 \text {, and } \phi=45^{\circ}

\end{aligned}