The projection of the vector \(5\hat{\imath}-\hat{\jmath}+3\hat{k}\) on the vector \(2\hat{\imath}+2\hat{\jmath}+\hat{k}\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{11}{3}\)
Let \(\vec{a}=5\hat{\imath}-\hat{\jmath}+3\hat{k}\) and \(\vec{b}=2\hat{\imath}+2\hat{\jmath}+\hat{k}\). The projection of \(\vec{a}\) on \(\vec{b}\) is \(\displaystyle\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}\).
\(\vec{a}\cdot\vec{b} = 10-2+3 = 11\)
\(|\vec{b}| = \sqrt{4+4+1} = 3\)
So the projection is
\( \displaystyle \frac{11}{3} \)
Why the other options are incorrect
Option (2) \(\displaystyle\frac{11}{\sqrt{35}}\) — Incorrect. It divides by \(|\vec{a}| = \sqrt{35}\); that is the projection of \(\vec{b}\) on \(\vec{a}\).
Option (3) \(11\) — Incorrect. It is \(\vec{a}\cdot\vec{b}\) itself, not divided by \(|\vec{b}| = 3\).
Option (4) \(\displaystyle\frac{11}{9}\) — Incorrect. It divides by \(|\vec{b}|^2 = 9\) instead of \(|\vec{b}| = 3\).