If \(\vec{a}=2\hat{\imath}+\hat{\jmath}+5\hat{k}\) and \(\vec{b}=\hat{\imath}+3\hat{\jmath}+7\hat{k}\), then the unit vector in the direction of \(\vec{a}+\vec{b}\) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \displaystyle \frac{1}{13}(3\hat{\imath}+4\hat{\jmath}+12\hat{k}) \)
Adding the components:
\(\vec{a}+\vec{b} = 3\hat{\imath}+4\hat{\jmath}+12\hat{k}\)
\(|\vec{a}+\vec{b}| = \sqrt{9+16+144} = \sqrt{169} = 13\)
The unit vector in a direction is the vector divided by its magnitude:
\( \displaystyle \frac{\vec{a}+\vec{b}}{|\vec{a}+\vec{b}|} = \frac{1}{13}(3\hat{\imath}+4\hat{\jmath}+12\hat{k}) \)