A vector \(\vec{r}\) has length 21 and its direction ratios are proportional to 2, −3, 6 then \(\vec{r}\) is equal to:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(6\vec{\imath}-9\vec{\jmath}+18\vec{k}\)
A vector of length 21 along a direction is 21 times the unit vector in that direction. The vector with the given direction ratios has length
\( \displaystyle \left|2\vec{\imath}-3\vec{\jmath}+6\vec{k}\right| = \sqrt{4+9+36} = 7 \)
so the unit vector is
\( \displaystyle \frac{2}{7}\vec{\imath}-\frac{3}{7}\vec{\jmath}+\frac{6}{7}\vec{k} \)
Hence
\( \displaystyle \vec{r} = 21\left(\frac{2}{7}\vec{\imath}-\frac{3}{7}\vec{\jmath}+\frac{6}{7}\vec{k}\right) \)
\(= 6\vec{\imath}-9\vec{\jmath}+18\vec{k}\)
(The opposite vector \(-6\vec{\imath}+9\vec{\jmath}-18\vec{k}\) also has length 21 and these direction ratios, but it is not among the options.)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{2}{7}\vec{\imath}-\frac{3}{7}\vec{\jmath}+\frac{6}{7}\vec{k}\) — Incorrect. This is the unit vector; its length is 1, not 21.
Option (3) \(2\vec{\imath}-3\vec{\jmath}+6\vec{k}\) — Incorrect. Its length is 7, not 21.
Option (4) \(21\left(2\vec{\imath}-3\vec{\jmath}+6\vec{k}\right)\) — Incorrect. Its length is \(21 \times 7 = 147\); the factor 21 must multiply the unit vector.