question_answer

Find a vector in the direction of a vector \[\vec{a}=-\,\hat{i}-\hat{j}+\hat{k},\] which has magnitude 8 units. \[\vec{a}=\hat{i}-\hat{j}+\hat{k},\] which has magnitude 8 units.

Answer:

Given, \[\vec{a}=\hat{i}-\hat{j}+\hat{k}\] Now, unit vector in the direction of \[\vec{a}\] is             \[\hat{a}=\frac{{\vec{a}}}{|\vec{a}|}=\frac{\hat{i}-\hat{j}+\hat{k}}{\sqrt{{{(1)}^{2}}+{{(-\,1)}^{2}}+(1)}}=\frac{\hat{i}-\hat{j}+\hat{k}}{\sqrt{3}}\] \[\therefore \]  Vector of magnitude 8 units in direction of \[\hat{a}\]             \[=\frac{8\,(\hat{i}-\hat{j}+\hat{k})}{\sqrt{3}}=\frac{8}{\sqrt{3}}\hat{i}-\frac{8}{\sqrt{3}}\hat{j}+\frac{8}{\sqrt{3}}\hat{k}\]