Question

If the vectors \[\vec{a} = 2 \hat{i} - 3 \hat{j}\] and \[\vec{b} = - 6 \hat{i} + m \hat{j}\] are collinear, find the value of m.

Answer 1

It is given that the vectors \[\vec{a} = 2 \hat{i} - 3 \hat{j}\] and \[\vec{b} = - 6 \hat{i} + m \hat{j}\] are collinear
\[\therefore \vec{b} = \lambda \vec{a}\]  for some scalar λ
\[\Rightarrow - 6 \hat{i} + m \hat{j} = \lambda\left( 2 \hat{i} - 3 \hat{j} \right)\]
\[ \Rightarrow - 6 \hat{i} + m \hat{j} = 2\lambda \hat{i} - 3\lambda \hat{j} \]
\[ \Rightarrow - 6 = 2\lambda\text{ and }m = - 3\lambda\]
\[ \Rightarrow m = - 3 \times \left( \frac{- 6}{2} \right) = 9\]
Thus, the value of m is 9.