The projection of the vector \(2\hat{\imath}-\hat{\jmath}+3\hat{k}\) on the vector \(3\hat{\imath} +2\hat{\jmath}+6\hat{k}\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{22}{7}\)
The projection of \(\vec a\) on \(\vec b\) is \(\frac{\vec a \cdot \vec b}{|\vec b|}\). With \(\vec a = 2\hat{\imath} - \hat{\jmath} + 3\hat{k}\) and \(\vec b = 3\hat{\imath} + 2\hat{\jmath} + 6\hat{k}\):
\(\vec a \cdot \vec b = 6 - 2 + 18 = 22\)
\(|\vec b| = \sqrt{9 + 4 + 36} = \sqrt{49} = 7\)
\(\displaystyle \text{Projection} = \frac{22}{7}\)
Why the other options are incorrect
Option (2) \(\displaystyle\frac{26}{49}\) — Incorrect. The dot product is 22, not 26, and the projection divides by \(|\vec b| = 7\).
Option (3) \(\displaystyle\frac{22}{49}\) — Incorrect. This divides by \(|\vec b|^2 = 49\). That number is the multiplier in the projection vector \(\frac{\vec a \cdot \vec b}{|\vec b|^2}\,\vec b\), not the projection.
Option (4) \(\displaystyle\frac{-22}{7}\) — Incorrect. \(\vec a \cdot \vec b = 22\) is positive, so the projection is positive.