The projection of the vector \(5\hat{\imath} + \hat{\jmath} - 3\hat{k}\) on the vector \(\hat{\imath} + 2\hat{\jmath} - 3\hat{k}\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{16}{\sqrt{14}}\)
Let \(\vec{a} = 5\hat{\imath} + \hat{\jmath} - 3\hat{k}\) and \(\vec{b} = \hat{\imath} + 2\hat{\jmath} - 3\hat{k}\). The projection of \(\vec{a}\) on \(\vec{b}\) is
\( \displaystyle \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|} \)
\( \vec{a}\cdot\vec{b} = 5(1) + 1(2) + (-3)(-3) = 5 + 2 + 9 = 16 \)
\( |\vec{b}| = \sqrt{1 + 4 + 9} = \sqrt{14} \)
So the projection is
\( \displaystyle \frac{16}{\sqrt{14}} \)
Why the other options are incorrect
Option (2) \(\displaystyle\frac{8}{7}\) — Incorrect. This is \(\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|^2} = \frac{16}{14}\), the multiplier of \(\vec{b}\) in the projection vector, not the projection itself.
Option (3) \(\displaystyle\frac{15}{13}\) — Incorrect. It does not equal \(\frac{16}{\sqrt{14}}\).
Option (4) \(\displaystyle\frac{16}{\sqrt{35}}\) — Incorrect. This divides by \(|\vec{a}| = \sqrt{25 + 1 + 9} = \sqrt{35}\). The projection on \(\vec{b}\) divides by \(|\vec{b}|\).