Question

If \[\vec{a}\text{ and }\vec{b}\] are two collinear vectors, then which of the following are incorrect?

Options

• \[\vec{b} = \lambda \vec{a}\] for some scalar λ

• \[\vec{a} = \pm \vec{b}\]

• the respective components of \[\vec{a}\text{ and }\vec{b}\] are proportional

• both the vectors \[\vec{a}\text{ and }\vec{b}\] have the same direction but different magnitudes

   

   

Answer 1

both the vectors \[\vec{a}\text{ and }\vec{b}\] have the same direction but different magnitudes
If \[\vec{a}\text{ and }\vec{b}\] are collinear vectors, then they are paprallel. Therefore, we have \[\vec{b} = \lambda \vec{a}\]  , for some scalar \[\lambda\]
If \[\lambda = \pm 1\]
\[\Rightarrow \vec{a} = \pm \vec{b} .\]
If \[b = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k}\] and \[\vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}\].
Then,
\[\vec{b} = \lambda \vec{a} . \]
\[ \Rightarrow b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} = \lambda \left( a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \right) . \]
\[ \Rightarrow b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} = \left( \lambda a_1 \right) \hat{i} + \left( \lambda a_2 \right) \hat{j} + \left( \lambda a_3 \right) \hat{k} . \]
\[ \Rightarrow b_1 = \lambda a_1 , b_2 = \lambda a_2 , b_3 = \lambda a_3 . \]
\[ \Rightarrow \frac{b_1}{a_1} = \frac{b_2}{a_2} = \frac{b_3}{a_3} = \lambda .\]
Thus, the respective components of \[\vec{a}\text{ and }\vec{b}\]  can have different directions. Hence, the statement given in (d) is incorrect.