Fundamentals of Physics/Vectors

< Fundamentals of Physics

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:

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 subtractionEdit

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)


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})