The projections of a vector on the x-axis, y-axis and z-axis are 4, −4 and 2 respectively. The direction cosines of the vector are
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{2}{3}, -\frac{2}{3}, \frac{1}{3}\)
The projections of a vector on the axes are its components, so the vector is
\(\vec{a} = 4\hat{\imath}-4\hat{\jmath}+2\hat{k}\)
Its magnitude:
\(|\vec{a}| = \sqrt{16+16+4} = \sqrt{36} = 6\)
The direction cosines are the components divided by the magnitude:
\( \displaystyle \frac{4}{6}, \frac{-4}{6}, \frac{2}{6} = \frac{2}{3}, -\frac{2}{3}, \frac{1}{3} \)
Why the other options are incorrect
Option (2) 4, −4, 2 — Incorrect. These are the direction ratios; their squares add to 36, not 1.
Option (3) \(\displaystyle -\frac{2}{3}, \frac{2}{3}, -\frac{1}{3}\) — Incorrect. These belong to the opposite vector \(-\vec{a}\); a vector has one fixed direction.
Option (4) \(\displaystyle\frac{1}{3}, -\frac{1}{3}, \frac{1}{6}\) — Incorrect. This divides by 12 instead of 6; the squares add to \(\frac{1}{4}\), not 1.