Fundamentals of Physics/Vectors

A vector is a two-element value that represents both magnitude and direction.

Vectors are normally represented by the ordered pair ${v} = (v_x\, v_y)$ or, when dealing with three dimentions, the tuple ${v} = (v_x\, v_y\, v_z)$. When written in this fashion, they represent a quantity along a given axis.

The following formulas are important with vectors:

$\left\|\mathbf{v}\right\|=\sqrt{{v_x}^2+{v_y}^2+{v_z}^2}$
$v_x = \left\|\mathbf{v}\right\| \cos{\theta}$
$v_y = \left\|\mathbf{v}\right\| \sin{\theta}$
$\theta = \tan^{-1}(\frac{v_y}{v_x})\,\!$

Addition is performed by adding the components of the vector. For example, c = a + b is seen as:

${c} = (a_x + b_x \, a_y + b_y)$

With subtraction, invert the sign of the second vector's components.

${c} = (a_x - b_x \, a_y - b_y)$

Multiplication (Scalar)Edit

The components of the vector are multiplied by the scalar:

$s * {v} = (s*v_x \, s*v_y)$

DivisionEdit

While some domains may permit division of vectors by vectors, such operations in physics are undefined. It is only possible to divide a vector by a scalar.