A vector of magnitude 10 units in the direction opposite to that of the vector \(3\hat{\imath}-4\hat{k}\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(-6\hat{\imath}+8\hat{k}\)
Let \(\vec{a}=3\hat{\imath}-4\hat{k}\). Its magnitude is
\(|\vec{a}| = \sqrt{3^2+(-4)^2} = 5\)
The unit vector along \(\vec{a}\) is
\( \displaystyle \hat{a} = \frac{3\hat{\imath}-4\hat{k}}{5} \)
A vector of magnitude 10 in the opposite direction is \(-10\,\hat{a}\):
\( \displaystyle -10\times\frac{3\hat{\imath}-4\hat{k}}{5} = -2(3\hat{\imath}-4\hat{k}) \)
\( = -6\hat{\imath}+8\hat{k}\)
Why the other options are incorrect
Option (2) \(6\hat{\imath}-8\hat{k}\) — Incorrect. It has magnitude 10 but points in the same direction as \(3\hat{\imath}-4\hat{k}\).
Option (3) \(-30\hat{\imath}+40\hat{k}\) — Incorrect. Its magnitude is \(\sqrt{900+1600} = 50\), not 10.
Option (4) \(\displaystyle\frac{-3\hat{\imath}+4\hat{k}}{5}\) — Incorrect. It is a unit vector, of magnitude 1.