Question

Let $\overrightarrow{X}$ and $\overrightarrow{Y}$ be the components of $\overrightarrow{P}$ along $x-$ axis and $y-$axis respectively, then, which of the following statements is/are true?

$\left(A\right)$ Magnitudes of both the components may be equal.

$\left(B\right)$ Magnitude of component $(X$ or $Y)$ may be equal to the magnitude of $\overrightarrow{P}$.

$\left(C\right)$ Magnitude of component $(X$ or $Y)$ may be greater than the magnitude of $\overrightarrow{P}$.

$\left(D\right)$ Magnitude of both the components may be equal to the magnitude of $\overrightarrow{P}$.

• A. All the statements are true.
• B. Statements $A$ and $B$ are true.
• C. Statements $B$ and $C$ are true.
• D. Statements $C$ and $D$ are true.

Answer 1

The correct option is B Statements $A$ and $B$ are true.

If $\overrightarrow{P}$ makes an angle of ${45}^{\circ }$ with either of the axes, the magnitudes of $\overrightarrow{X}$ and $\overrightarrow{Y}$ will be equal.

The statement $A$ is true.

If $\overrightarrow{P}$ is directed along $x-$axis, then the magnitude of $\overrightarrow{X}$ will be equal to the magnitude of $\overrightarrow{P}$ and magnitude of $\overrightarrow{Y}$ will be zero.

Similarly, if $\overrightarrow{P}$ is directed along $y-$axis, then the magnitude of $\overrightarrow{Y}$ will be equal to the magnitude of $\overrightarrow{P}$ and magnitude of $\overrightarrow{X}$ will be zero.

The statement $B$ is true.

Also, magnitude of both the components can never be equal to the magnitude of $\overrightarrow{P}$.

The statement $D$ is false.

The magnitude of any component $(X$ or $Y)$ of a vector is never greater than the magnitude of the vector itself.

The statement $C$ is false.

Hence, option $\left(B\right)$ is the correct answer.