Question

If \(\vec x\) and \(\vec y\) are two collinear vectors, then which of the following are incorrect?

• A. \(\vec x=±\vec y\)
• B. \(\vec y=λ\vec x\), for some scalar \(λ\)
• C. Both the vectors \(\vec x\) and \(\vec y\) have same direction, but different magnitudes.
• D. the respective components of \(\vec x\) and \(\vec y\) are not proportional

Answer 1

The Correct Option is D

Two vectors \( \vec x \) and \( \vec y \) are collinear if they lie along the same line. This implies that one vector is a scalar multiple of the other. Therefore, for collinear vectors:

1. \( \vec x = ±\vec y \): This statement is correct because if vectors have the same or opposite directions, it means they are collinear.
2. \( \vec y = λ \vec x \): This statement is correct as one vector can be expressed as a scalar multiple of the other for collinearity, where \( λ \) is any scalar.
3. Both vectors \( \vec x \) and \( \vec y \) have the same direction, but different magnitudes: This statement can also be correct if we assume the scalar \( λ \) to be positive. However, it does not encompass all collinearity cases.
4. The respective components of \( \vec x \) and \( \vec y \) are not proportional: This is incorrect as by definition collinear vectors have proportional components.

Statement about components:

The respective components of \( \vec x \) and \( \vec y \) are not proportional

Conclusion: The statement about the components not being proportional is incorrect as it contradicts the property of collinear vectors. Hence, the incorrect option is when it's stated that the respective components of \( \vec x \) and \( \vec y \) are not proportional.