Question

Find a vector of magnitude 5units, and parallel to the resultant of the vectors $\vec{a}=2\hat{i}+3\hat{j}-\hat{k}\space and\space \vec{b}=\hat{i}-2\hat{j}+\hat{k}.$

Answer 1

We have,
$\vec{a}=2\hat{i}+3\hat{j}-\hat{k}\space and\space \vec{b}=\hat{i}-2\hat{j}+\hat{k}.$
Let $\vec{c}$ be the resultant of a→and b→.
Then,
$\vec{c}=\vec{a}+\vec{b}=(2+1)\hat{i}+(3-2)\hat{j}+(-1+1)\hat{k}=3\hat{i}+\hat{j}$
$∴|\vec{c}|=\sqrt{3^{2}+1^{2}}\sqrt{9+1}=\sqrt{10}$
$∴\hat{c}=\frac{\vec{c}}{|\vec{c}|}=\frac{(3\hat{i}+\hat{j})}{\sqrt{10}}$
Hence,the vector of magnitude 5units and parallel to the resultant of vectors $\vec{a}$ and $\vec{b}$ is \pm5.\hat{c}=$$\pm5.\frac{1}{\sqrt{10}}(3\hat{i}+\hat{j})$$=\pm\frac{3\sqrt{10}\hat{i}}{2}\pm\frac{\sqrt{10}}{2}\hat{j}.