Definitions

Orthogonal
Two vectors are orthogonal if their dot product is equal to zero by the relationship

(1)
\begin{align} \vec{u} \cdot \vec{v} = |\vec{u}||\vec{v}| \cos{\theta} \end{align}

If two vectors are orthogonal (90 degrees or $\frac{\pi}{2}$ rads), $\cos{\theta}=0$.

Unit Vector
A vector of magnitude 1. This is often denoted as $\hat{v}$. In fact, in physics, this is where we get the symbols $\hat{i} \hat{j} \text{ and } \hat{k}$ which symbolize the unit vectors in the x, y, and z directions respectively.

To make a vector a unit vector, simply divide each element by its magnitude which is written as $\frac{\vec{r}}{|\vec{r}|} = \hat{r}$

On a side note, if two vectors are orthogonal, $|\vec{u}|^2 + |\vec{v}|^2 = |\vec{u+v}|^2$ because, if $\vec{u}$ and $\vec{v}$ are orthogonal, they intersect at a right angle and $\vec{u+v}$ signifies the diagonal of the parallelogram formed by the two vectors. In the case of two orthogonal vectors, the parallelogram is a rectangle (remember squares are rectangles, but rectangles are not necessarily squares) and $\vec{u+v}$ is the hypotenuse of a right triangle. Thus by the Pythagorean theorem, $|\vec{u}|^2 + |\vec{v}|^2 = |\vec{u+v}|^2$.

page revision: 0, last edited: 26 Apr 2015 20:50