The values of \(\lambda\) and \(\mu\) for which \((3\hat{\imath}-\lambda\hat{\jmath}+6\hat{k})\times(\mu\hat{\imath}+2\hat{\jmath}-4\hat{k}) = \vec{0}\) are
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\lambda = 3,\ \mu = -2\)
The cross product of two non-zero vectors is \(\vec{0}\) exactly when the vectors are parallel, that is, when their components are proportional:
\( \displaystyle \frac{3}{\mu} = \frac{-\lambda}{2} = \frac{6}{-4} \)
The last ratio is \(\displaystyle -\frac{3}{2}\). From the first ratio:
\( \displaystyle \frac{3}{\mu} = -\frac{3}{2} \)
\(\mu = -2\)
From the second ratio:
\( \displaystyle \frac{-\lambda}{2} = -\frac{3}{2} \)
\(\lambda = 3\)