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
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.
The correct answer is D.
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