Consider a vector $\vec{A}$ with three mutually perpendicular components $A_x,$ $A_y$ and $A_z$ along X, Y and Z-axis respectively. Then, $\vec{A}$ can be represented as,
\[\vec{A}=A_x\hat{i}+A_y\hat{j}+A_z\hat{k}\]
Its magnitude is given by, \[A=\sqrt{A_x^2+A_y^2+A_z^2}\]
Now, if the magnitude of the vector is doubled, then,
\[2A=2\sqrt{A_x^2+A_y^2+A_z^2}\] \[=\sqrt{4(A_x^2+A_y^2+A_z^2)}\] \[=\sqrt{(2A_x)^2+(2A_y)^2+(2A_z)^2}\]
Therefore, if the magnitude of a vector is doubled keeping the direction same, the magnitude of each component of the vector is doubled.