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 and subtraction

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)

The components of the vector are multiplied by the scalar:

$s*{v}=(s*v_{x}\,s*v_{y})$

Division

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.

As with multiplication, the components of the vector are divided by the scalar:

$s*{v}=({\frac {s_{x}}{v_{x}}}\,{\frac {s_{y}}{v_{y}}})$