Find a vector of magnitude 5 units, and parallel to the resultant of the vectors

We have,

$\vec{a}=2 \hat{i}+3 \hat{j}-\hat{k}$ and $\vec{b}=\hat{i}-2 \hat{j}+\hat{k}$

Then,

$\vec{c}=\vec{a}+\vec{b}=(2+1) \hat{i}+(3-2) \hat{j}+(-1+1) \hat{k}=3 \hat{i}+\hat{j}$

$\therefore|\vec{c}|=\sqrt{3^{2}+1^{2}}=\sqrt{9+1}=\sqrt{10}$

$\therefore \hat{c}=\frac{\vec{c}}{|\vec{c}|}=\frac{(3 \hat{i}+\hat{j})}{\sqrt{10}}$

Hence, the vector of magnitude 5 units and parallel to the resultant of vectors $\vec{a}$ and $\vec{b}$ is

$\pm 5 \cdot \hat{c}=\pm 5 \cdot \frac{1}{\sqrt{10}}(3 \hat{i}+\hat{j})=\pm \frac{3 \sqrt{10} \hat{i}}{2} \pm \frac{\sqrt{10}}{2} \hat{j}$