Directed components and magnitude

A vector has magnitude and direction. The column vector \(\begin{pmatrix}x\\ y\end{pmatrix}\) describes a displacement of \(x\) units horizontally and \(y\) units vertically. Right and up are positive; left and down are negative.

Here \(\overrightarrow{AB}=\overrightarrow{CD}\): the starting points differ, but the displacements have the same components. Reversing a vector gives its negative.

Add or subtract corresponding components. Multiplication by a positive scalar scales the length; multiplication by a negative scalar also reverses the direction. For a nonzero vector \(\mathbf{u}\), \(|k\mathbf{u}|=|k|\,|\mathbf{u}|\).

Use Pythagoras’ theorem for magnitude: \(\left|\begin{pmatrix}x\\ y\end{pmatrix}\right|=\sqrt{x^2+y^2}\). Magnitude is a nonnegative number, not a column vector. A translation adds the same vector to every point.