Discussion
If \(\vec{a} = 4\hat{i} + \lambda\hat{j} - 6\hat{k}\) and \(\vec{b} = -6\hat{i} + 12\hat{j} + 9\hat{k}\) are collinear, then the value of \(\lambda\) is equal to
Answer: D
\(\lambda = -8\).
Collinear vectors are scalar multiples, so their corresponding components are in the same ratio:
\[ \begin{aligned} \frac{4}{-6} &= \frac{\lambda}{12} \\ &= \frac{-6}{9} \\ &= -\frac{2}{3} \end{aligned} \]
\[ \lambda = 12 \times \left(-\tfrac{2}{3}\right) = -8 \]
The i- and k-components agree on the ratio \(-\frac{2}{3}\), which confirms the vectors can be collinear. The option 8 drops the minus sign.
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