If are two collinear vectors, then which of the following are incorrect:
Question:

If $\vec{a}$ and $\vec{b}$ are two collinear vectors, then which of the following are incorrect:

A. $\vec{b}=\lambda \vec{a}$, for some scalar $\lambda$

B. $\vec{a}=\pm \vec{b}$

C. the respective components of $\vec{a}$ and $\vec{b}$ are proportional

D. both the vectors $\vec{a}$ and $\vec{b}$ have same direction, but different magnitudes

Solution:

If $\vec{a}$ and $\vec{b}$ are two collinear vectors, then they are parallel.

Therefore, we have:

$\vec{b}=\lambda \vec{a}$ (For some scalar $\lambda$ )

If $\lambda=\pm 1$, then $\vec{a}=\pm \vec{b}$.

If $\vec{a}=a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k}$ and $\vec{b}=b_{1} \hat{i}+b_{2} \hat{j}+b_{3} \hat{k}$, 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 $\vec{a}$ and $\vec{b}$ are proportional.

However, vectors $\vec{a}$ and $\vec{b}$ can have different directions.

Hence, the statement given in D is incorrect.