# The magnitude of a vector has doubled, its direction remaining the same. Can you conclude that the magnitude of each component of the vector has doubled?

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.