A vector can be defined by giving its magnitude, and the angle between the vector and one of the coordinate axes. This is known as the magnitude-direction form.

The magnitude of a vector can be found by using Pythagoras' Theorem. The notation for magnitude is the same as the notation for modulus.

The angle between the vector and one of the coordinate axes can be found using trigonometry.

Magnitude of a 2D vector

\(\bold{a} = x\bold{i} + y\bold{j} = \begin{pmatrix} x \\ y \end{pmatrix} \)

\(\implies \boxed{|\bold{a}| = \sqrt{x^2+y^2}} \)

Magnitude of a 3D vector

\(\bold{a} = x\bold{i} + y\bold{j} + z\bold{k} = \begin{pmatrix} x \\ y \\ z \end{pmatrix} \)

\(\implies \boxed{|\bold{a}| = \sqrt{x^2+y^2+z^2}} \)

Direction of a 2D vector

\(\bold{a} = x\bold{i} + y\bold{j} = \begin{pmatrix} x \\ y \end{pmatrix} \)

Angle \(\theta_x\) between the vector and the \(x\)-axis:

\(\theta_x = \arctan{\dfrac{y}{x}} \)

Angle \(\theta_y\) between the vector and the \(y\)-axis:

\(\theta_y = \arctan{\dfrac{x}{y}} \)

Direction of a 3D vector

\(\bold{a} = x\bold{i} + y\bold{j} + z\bold{k} = \begin{pmatrix} x \\ y \\ z \end{pmatrix} \)

If the vector makes an angle \(\theta_x\) with the \(x\)-axis:

\(\theta_x = \arccos{\dfrac{x}{|a|}} \)

This also works for the \(y\)-axis and \(z\)-axis, and also for 2D vectors.

Finding unit vectors

A unit vector is any vector with a magnitude of \(1\).

Therefore a unit vector in the direction of \(\bold{a}\) is \(\dfrac{\bold{a}}{|\bold{a}|} \).

Parallel vectors

Any vector parallel to the vector \(\bold{a}\) may be written as \(\lambda\bold{a}\), where \(\lambda\) is a non-zero scalar.