The vector in the direction of the vector \(2\hat{\imath}-\hat{\jmath}-2\hat{k}\) that has magnitude 9 units is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \((6\hat{\imath}-3\hat{\jmath}-6\hat{k})\)
A vector of magnitude 9 in the direction of \(\vec a\) is \(9\hat a\), where \(\hat a\) is the unit vector along \(\vec a\).
\(\vec a = 2\hat{\imath} - \hat{\jmath} - 2\hat{k}\)
\(|\vec a| = \sqrt{4 + 1 + 4} = 3\)
\(\displaystyle \hat a = \frac{\vec a}{|\vec a|} = \frac{1}{3}(2\hat{\imath} - \hat{\jmath} - 2\hat{k})\)
\(9\hat a = 3(2\hat{\imath} - \hat{\jmath} - 2\hat{k})\)
\(= 6\hat{\imath} - 3\hat{\jmath} - 6\hat{k}\)
Why the other options are incorrect
Option (1) \(2\hat{\imath}-\hat{\jmath}-2\hat{k}\) — Incorrect. This is the given vector itself, of magnitude 3.
Option (2) \(9(2\hat{\imath}-\hat{\jmath}-2\hat{k})\) — Incorrect. Its magnitude is \(9 \times 3 = 27\). The factor 9 must multiply the unit vector, not the given vector.
Option (3) \(\displaystyle\frac{1}{9}(2\hat{\imath}-\hat{\jmath}-2\hat{k})\) — Incorrect. Its magnitude is \(\frac{3}{9} = \frac{1}{3}\).