Question

If $a$ and $b$ are two collinear vectors, then which of the following is/are incorrect?
I. $b=\lambda a$, for some scalar $\lambda ,\lambda \ne 0$.
II. $\hat{a}=\pm \hat{b}$
III. The respective components of $a$ and $b$ are not proportional.
IV. Both the vector $a$ and $b$ have same direction, but different magnitude.

• A. Only I is incorrect
• B. Both III and IV are incorrect
• C. Only III is incorrect
• D. Both I and II are incorrect

Answer 1

Answer: (B)

Considering each statement
Statement I b $=\lambda a,\lambda \ne 0$ is correct, it is the condition for collinearity of two vectors.
Statement II $\hat{a}=\pm \hat{b}$ is also correct, it shows that unit vectors in the direction of given vectors are either in the same direction or in opposite. $(\because a$ and $b$ are collinear) Statement III is incorrect, if for two vectors the respective components are proportional, then their magnitude will be different but they will be collinear.
Statement IV is incorrect, because if the two vectors are collinear then they may have same directions or opposite directions.