If \( \vec{a}=\hat{\imath}+\hat{\jmath}+2\hat{k} \) and \( \vec{b}=2\hat{\imath}+\hat{\jmath}+2\hat{k} \), then the unit vector in the direction of \( 2\vec{a}-\vec{b} \) is equal to :
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \displaystyle\frac{1}{\sqrt{5}}(\hat{\jmath}+2\hat{k}) \)
\( 2\vec{a}-\vec{b}=(2-2)\hat{\imath}+(2-1)\hat{\jmath}+(4-2)\hat{k} \)
\( =\hat{\jmath}+2\hat{k} \)
\( \displaystyle |2\vec{a}-\vec{b}|=\sqrt{0^{2}+1^{2}+2^{2}} \)
\( \displaystyle =\sqrt{5} \)
The unit vector is the vector divided by its magnitude:
\( \displaystyle \frac{2\vec{a}-\vec{b}}{|2\vec{a}-\vec{b}|}=\frac{1}{\sqrt{5}}(\hat{\jmath}+2\hat{k}) \)