If \(|\vec{a}| = 4\), \(|\vec{b}| = 3\) and the angle between \(\vec{a}\) and \(\vec{b}\) is \(\displaystyle\frac{2\pi}{3}\), then the projection of \(\vec{a}\) on \(\vec{b}\) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → −2
The projection of \(\vec{a}\) on \(\vec{b}\) is \(\displaystyle\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}\), which is the same as \(|\vec{a}|\cos\theta\):
\( \displaystyle \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|} = \frac{|\vec{a}|\,|\vec{b}|\cos\theta}{|\vec{b}|} = |\vec{a}|\cos\theta \)
\( \displaystyle = 4\cos\frac{2\pi}{3} \)
\( \displaystyle = 4\times\left(-\frac{1}{2}\right) = -2 \)
The projection is negative because the angle between the vectors is obtuse.
Why the other options are incorrect
Option (1) 2 — Incorrect. This drops the minus sign of \(\cos\displaystyle\frac{2\pi}{3}\).
Option (2) −6 — Incorrect. This is \(\vec{a}\cdot\vec{b} = 4\times3\times\left(-\displaystyle\frac{1}{2}\right)\), not yet divided by \(|\vec{b}|\).
Option (3) \(-\displaystyle\frac{3}{2}\) — Incorrect. This is the projection of \(\vec{b}\) on \(\vec{a}\), \(\displaystyle\frac{-6}{4}\).