Question

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

• A. $\vec{b}=λ\vec{a}$,for some scalar $λ$
• B. $\vec{a}=±\vec{b}$
• 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

Answer 1

The Correct Option is D

The correct answer is 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:
$\vec{b}=λ\vec{a}$(for some scalar $λ$)
If $λ=±1$,then $\vec{a}=±\vec{b}.$
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
$\vec{b}=λ\vec{a}.$
$\implies b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$$=\lambda(a_1\hat{i}+a_2\hat{j}+a_3\hat{k})$
$\implies b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$$=(λa_1)\hat{i}+(λa_2)\hat{j}+(λa_3)\hat{k}$
$⇒b_1=λa_1,b_2=λa_2,b_3=λa_3$
$⇒\frac{b_1}{a_1}=\frac{b_2}{a_2}=\frac{b_3}{a_3}=λ$
Therefore,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.