Question

The component of a vector $r$ along the $X$-axis will have maximum value if:

  • $r$ is along positive $X$-axis

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Answer

To solve the problem of determining when the component of a vector $\mathbf{r}$ along the $X$-axis will have the maximum value, let's consider the options given:

  1. $\mathbf{r}$ is along the positive $Y$-axis.
  2. $\mathbf{r}$ is along the positive $X$-axis.
  3. $\mathbf{r}$ makes an angle of $45^\circ$ with the $Y$-axis.
  4. $\mathbf{r}$ is along the negative $Y$-axis.

Let's analyze the situation:

  1. When a vector $\mathbf{r}$ makes an angle $\theta$ with the $X$-axis, the component of $\mathbf{r}$ along the $X$-axis is given by: $$ r_x = r \cos(\theta) $$ where $r$ is the magnitude of the vector $\mathbf{r}$ and $\theta$ is the angle between $\mathbf{r}$ and the $X$-axis.

  2. To maximize $r_x$, we need to maximize $\cos(\theta)$. The maximum value of $\cos(\theta)$ is $1$, which occurs when $\theta = 0^\circ$. This means that the vector $\mathbf{r}$ should be aligned with the $X$-axis.

Now, let's examine each option:

  1. $\mathbf{r}$ is along the positive $Y$-axis: In this situation, $\mathbf{r}$ is perpendicular to the $X$-axis, i.e., $\theta = 90^\circ$ and $\cos(90^\circ) = 0$. Thus, $r_x = 0$, which is not the maximum value.

  2. $\mathbf{r}$ is along the positive $X$-axis: Here, $\theta = 0^\circ$ and $\cos(0^\circ) = 1$. Therefore, $r_x = r \cdot 1 = r$, which is the maximum possible value for the component along the $X$-axis.

  3. $\mathbf{r}$ makes an angle of $45^\circ$ with the $Y$-axis: This implies that $\mathbf{r}$ makes an angle $\theta = 45^\circ$ with the $X$-axis. The cosine of $45^\circ$ is $\cos(45^\circ) = \frac{1}{\sqrt{2}}$. Thus, $r_x = r \cos(45^\circ) = \frac{r}{\sqrt{2}}$, which is less than $r$.

  4. $\mathbf{r}$ is along the negative $Y$-axis: In this case, $\theta = 270^\circ$ (or $-90^\circ$). Here also, $\cos(270^\circ) = 0$. Thus, $r_x = 0$, which is not the maximum value.

From this analysis, it is evident that the maximum component of the vector $\mathbf{r}$ along the $X$-axis occurs when $\mathbf{r}$ is along the positive $X$-axis.

Therefore, the correct option is:

  • (B) $\mathbf{r}$ is along the positive $X$-axis.

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