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:

$\Vert 𝐯\Vert =\sqrt{{v_x}^2+{v_y}^2+{v_z}^2}$

$v_x=\Vert 𝐯\Vert \cos \theta$

$v_y=\Vert 𝐯\Vert \sin \theta$

$\theta ={\tan }^{-1}(\frac{v_y}{v_x})$

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:

$s*v=(\frac{s_x}{v_x}\frac{s_y}{v_y})$

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