Question

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

Answer 1

Given vectors $a=2\hat{i}+3\hat{j}-\hat{k}andb=\hat{i}-2\hat{j}+\hat{k}.$
Let $\overrightarrow{c}$ be the resultant of a and b.
$\therefore c=a+b=(2\hat{i}+3\hat{j}-\hat{k})+(\hat{i}-2\hat{j}+\hat{k})\Rightarrow c=3\hat{i}+\hat{j}+0\hat{k}$
Comparing with $X=x\hat{i}+y\hat{j}+z\hat{k}$
$\therefore |c|=\sqrt{x^2+y^2+z^2}=\sqrt{3^2+1^2}=\sqrt{9+1}=\sqrt{10}$
$\therefore$ Unit vector in the direction of c, $\hat{c}=\frac{c}{|c|}=\frac{3\hat{i}+\hat{j}}{\sqrt{10}}$
Hence, the vector of magnitude 5 units and parallel to the resultant of vectors a and b is $\pm 5\hat{c}=\pm 5\frac{1}{\sqrt{10}}(3\hat{i}+\hat{j})=\pm \frac{3\sqrt{10}}{2}\hat{i}\pm \frac{\sqrt{10}}{2}\hat{j}$