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

The correct option is r is along positive X-axis.

To understand why, let us analyze the component of a vector along a given axis. Consider a vector
$\mathbf{r}$
of magnitude
$r$
that makes an angle
$\theta$
with the positive X-axis. The component of this vector along the X-axis, denoted as
$r_x$
is given by the formula:
$r_x=rcos\left(\theta \right)$
where
$r\ge 0$
is the magnitude of the vector.

To find the condition under which the X-component
$r_x$
has its maximum value, we need to maximize the term
$cos\left(\theta \right)$
.

The cosine function,
$cos\left(\theta \right)$
oscillates between a minimum value of -1 and a maximum value of +1:
$-1\le cos\left(\theta \right)\le 1$
Thus, the maximum value of
$cos\left(\theta \right)$
is 1.

This maximum value of 1 occurs when the angle
$\theta$
is:
$\theta =0^{\circ }$
An angle of
$0^{\circ }$
with the X-axis means that the vector
$\mathbf{r}$
points directly along the positive X-axis. In this case, the entire magnitude of the vector lies along the X-axis, giving
$r_x=r$
.