The projection vector of the vector \(2\hat{\imath} + 3\hat{\jmath} + \hat{k}\) on \(2\hat{\imath} + \hat{\jmath} - 2\hat{k}\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{5}{9}(2\hat{\imath} + \hat{\jmath} - 2\hat{k})\)
Let \(\vec{a} = 2\hat{\imath} + 3\hat{\jmath} + \hat{k}\) and \(\vec{b} = 2\hat{\imath} + \hat{\jmath} - 2\hat{k}\). The projection vector of \(\vec{a}\) on \(\vec{b}\) is
\(\displaystyle \left(\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|^2}\right)\vec{b}\)
\(\vec{a}\cdot\vec{b} = 4 + 3 - 2 = 5\)
\(|\vec{b}|^2 = 4 + 1 + 4 = 9\)
So the projection vector is
\(\displaystyle \frac{5}{9}(2\hat{\imath} + \hat{\jmath} - 2\hat{k})\)
Why the other options are incorrect
Option (1) \(\frac{5}{3}(2\hat{\imath} + \hat{\jmath} - 2\hat{k})\) — Incorrect. It divides by \(|\vec{b}| = 3\) instead of \(|\vec{b}|^2 = 9\).
Option (3) \(\frac{5}{14}(2\hat{\imath} + 3\hat{\jmath} + \hat{k})\) — Incorrect. This is the projection of \(\vec{b}\) on \(\vec{a}\), since \(|\vec{a}|^2 = 14\).
Option (4) \(\frac{5}{3}\) — Incorrect. \(\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|} = \frac{5}{3}\) is the scalar projection, a number, not the projection vector.