A component of a vector is

Let a vector be represented as $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$.

The magnitude of the vector is given by $A = \sqrt{A_x^2 + A_y^2 + A_z^2}$.

Since the squares of real numbers are non-negative, $A_y^2 \ge 0$ and $A_z^2 \ge 0$. This implies that $A_x^2 \le A_x^2 + A_y^2 + A_z^2$, which gives $A_x^2 \le A^2$.

Taking the square root on both sides, we get $|A_x| \le A$.

Similarly, $|A_y| \le A$ and $|A_z| \le A$.

Therefore, the magnitude of any component of a vector is always less than or equal to the magnitude of the vector itself. If the vector lies entirely along a single axis, its component along that axis is exactly equal to its magnitude.