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 bmagnitude-direction form/b.

The bmagnitude/b 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.

[b]uMagnitude of a 2D vector[/u]/b

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

(\implies \boxed{|\bold{a}| = \sqrt{x^2+y2}} )

[b]uMagnitude of a 3D vector[/u]/b

(\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+y2+z2}} )

[b]uDirection of a 2D vector[/u]/b

(\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}} )

[b]uDirection of a 3D vector[/u]/b

(\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.

[b]uFinding unit vectors[/u]/b

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}|} ).

[b]uParallel vectors[/u]/b

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

[b]uMagnitude and direction[/u]/b

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

(|\bold{a}| = \sqrt{x^2+y2+z2} )

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

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