Position vector of a point P(x, y, z) in space

For a point $P(x,y,z)$ in space (with respect to the origin $O(0,0,0)$), the vector $\overrightarrow{OP}$ is

called the position vector of $P$ with respect to $O$. Dropping perpendiculars from $P$ gives

$\overrightarrow{OA}=x\hat\imath,\ \overrightarrow{OB}=y\hat\jmath,\ \overrightarrow{OC}=z\hat k$ along the

three axes, and chaining the triangle law through $\triangle OLP$ and $\triangle OAL$ (using

$\overrightarrow{AL}=\overrightarrow{OB}$) gives

Squaring and adding via the same two right triangles ($OP^2=OL^2+LP^2=OA^2+AL^2+LP^2=OA^2+OB^2+OC^2$) gives …