Vector projection of \( 7\hat{\imath}+\hat{\jmath}+4\hat{k} \) on the vector \( 2\hat{\imath}+6\hat{\jmath}-3\hat{k} \) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{8}{49}(2\hat{\imath}+6\hat{\jmath}-3\hat{k}) \)
Let \( \vec{a}=7\hat{\imath}+\hat{\jmath}+4\hat{k} \) and \( \vec{b}=2\hat{\imath}+6\hat{\jmath}-3\hat{k} \). The vector projection of \( \vec{a} \) on \( \vec{b} \) is
\( \displaystyle \left(\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|^2}\right)\vec{b} \)
\( \displaystyle \vec{a}\cdot\vec{b}=14+6-12=8 \)
\( \displaystyle |\vec{b}|^2=4+36+9=49 \)
\( \displaystyle \text{Vector projection}=\frac{8}{49}(2\hat{\imath}+6\hat{\jmath}-3\hat{k}) \)
Why the other options are incorrect
Option (1) \( \displaystyle\frac{8}{7} \) — Incorrect. \( \frac{8}{7}=\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|} \) is the scalar projection (a number), not the vector projection.
Option (2) \( \displaystyle\frac{8}{7}(2\hat{\imath}+6\hat{\jmath}-3\hat{k}) \) — Incorrect. This multiplies the scalar projection by \( \vec{b} \) itself; it must multiply the unit vector \( \frac{\vec{b}}{7} \), which gives \( \frac{8}{49}\vec{b} \).
Option (4) \( \displaystyle\frac{8}{49}(7\hat{\imath}+\hat{\jmath}+4\hat{k}) \) — Incorrect. The vector projection on \( \vec{b} \) must be along \( \vec{b} \), not along \( \vec{a} \).