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:


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