Question

If $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$are two collinear vectors, then which of the following are incorrect?
(a) $\overrightarrow{b}=\lambda \overrightarrow{a}$ for some scalar λ
(b) $\overrightarrow{a}=\pm \overrightarrow{b}$
(c) the respective components of $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$ are proportional
(d) both the vectors $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$have the same direction but different magnitudes

Answer 1

(d) both the vectors $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$have the same direction but different magnitudes
If $\overrightarrow{a}$ and $\overrightarrow{b}$ are collinear vectors, then they are paprallel. Therefore, we have
$\overrightarrow{b}=\lambda \overrightarrow{a}$ , for some scalar $\lambda$.
If $\lambda =\pm 1$ $\Rightarrow \overrightarrow{a}=\pm \overrightarrow{b}.$
If $b=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ and $\overrightarrow{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$. Then,
$$\begin{gathered} \overrightarrow{b}=\lambda \overrightarrow{a}. \\ \Rightarrow b_1\hat{i}+b_2\hat{j}+b_3\hat{k}=\lambda \left(a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\right). \\ \Rightarrow b_1\hat{i}+b_2\hat{j}+b_3\hat{k}=\left(\lambda a_1\right)\hat{i}+\left(\lambda a_2\right)\hat{j}+\left(\lambda a_3\right)\hat{k}. \\ \Rightarrow b_1=\lambda a_1,b_2=\lambda a_2,b_3=\lambda a_3. \\ \Rightarrow \frac{b_1}{a_1}=\frac{b_2}{a_2}=\frac{b_3}{a_3}=\lambda . \end{gathered}$$

Thus, the respective components of $\overrightarrow{a}$ and $\overrightarrow{b}$ can have different directions. Hence, the statement given in (d) is incorrect.