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Q.
A unit vector in the direction of resultant vector of $\vec{A}=-2{\hat{i}}+3\hat{j}+\hat{k}$ and $\vec{B}=\hat{i}+2\hat{j}-4\hat{k}$ is
Motion in a Plane
Solution:
Here, $\vec{A}=-2\hat{\hat{i}}+3\hat{j}+\hat{k}$
$\vec{B}=\hat{i}+2\hat{j}-4\hat{k}$
The resultant vector of $\vec{A}$ and $\vec{B}$ is
$\vec{R}=\vec{A}+\vec{B}$
$\therefore \vec{R}=\left(-2\hat{i}+3\hat{j}+\hat{k}\right)+\left(\hat{i}+2\hat{j}-4\hat{k}\right)$
$=-\hat{i}+5\hat{j}-3\hat{k}$
$\left|\vec{R}\right|=\sqrt{\left(-1\right)^{2}+\left(5\right)^{2}+\left(-3\right)^{2}}$
$=\sqrt{1+25+9}$
$=\sqrt{35}$
Unit vector in the direction of resultant vector of $\vec{A}$ and $\vec{B}$ is
$\hat{R}=\frac{\vec{R}}{\left|\vec{R}\right|}$
$=\frac{-\hat{i}+5\hat{j}-3\hat{k}}{\sqrt{35}}$