The component of a vector r along the Xaxis will have maximum value if: r is along positive Xaxis
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:
 $\mathbf{r}$ is along the positive $Y$axis.
 $\mathbf{r}$ is along the positive $X$axis.
 $\mathbf{r}$ makes an angle of $45^\circ$ with the $Y$axis.
 $\mathbf{r}$ is along the negative $Y$axis.
Let's analyze the situation:

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.

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:

$\mathbf{r}$ 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.

$\mathbf{r}$ 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.

$\mathbf{r}$ 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 $r$.

$\mathbf{r}$ 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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