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.