Solve the following

Question:

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

Solution:

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

$B_{x}=-100 / \sqrt{2}$

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

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

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

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

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

$R^{2}=x^{2}+y^{2}=100^{2}$

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

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