Single Choice

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$$\vec {a}=\pm \vec {b}$$
Cthe respective components of $$\vec {a}$$ and $$\vec {b}$$ are proportional
Dboth the vectors $$\vec {a}$$ and $$\vec {b}$$ have same direction, but different magnitudes
Correct Answer

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
$$\vec {b}=\lambda \vec {a}$$
$$\Rightarrow b_1\hat {i}+b_2\hat {j}+b_3\hat {k}=\lambda (a_1\hat {i}+a_2\hat {j}+a_3\hat {k}=\lambda (a_1\hat {i}+a_2\hat {j}+a_3\hat {k})$$
$$\Rightarrow b_1\hat {i}+b_2\hat {j}+b_3\hat {k}=(\lambda a_1)\hat {i}+(\lambda a_2)\hat {j}+(\lambda a_3)\hat {k}$$
$$\Rightarrow b_1=\lambda a_1, b_1=\lambda a_2, b_3=\lambda a_3$$
$$\Rightarrow \dfrac {b_1}{a_1}=\dfrac {b_2}{a_2}=\dfrac {b_3}{a_3}=\lambda$$
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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