Euclidean Vectors

Euclidian Vectors are objects/measurements that have magnitude and direction in a space.  A vector is usually represented as $\vec{r}$ where $r$ is any letter.

Magnitude is equivalent to length of the vector - a quantity specifying how long it is from beginning to end when graphically represented in its respective coordinate system.  A magnitude is usually represented using double lines, $|\vec{r}|$, and is always positive.  It is a scalar, a singular value that is not a vector.

The direction is a separate vector that specifies the direction that a vector faces.  The direction of a vector is called a unit vector, and is represented as $\hat{r}$.  It is a different kind of vector that exists purely as a supplement.  For instance, in 3-dimensional space a magnitude could extend any which direction within that space because it is a single number.  It has no context, it is a number .  With the addition of a unit vector, it becomes a posited, measurable quantity that extends as far as the magnitude in the direction of the unit vector.

A vector $\vec{r}$ is the product of the magnitude $|\vec{r}|$ and unit vector $\hat{r}$:

$$\vec{r} = |\vec{r}|\times\hat{r}$$

Each vector is made up of components that each represent a dimension within the Euclidean space it resides.  For instance, in a Cartesian coordinate system with 3-dimensions X, Y, and Z:

 $$\vec{r} = <r_x, r_y, r_z>$$

To find the magnitude of a vector, square each component, sum them together, and then take the square root of the result.

$$|\vec{r}| = \sqrt{r_x^2 + r_y^2 + r_z^2}$$

To find the unit vector (direction) of a vector, divide the vector by its magnitude.  Each component will be divided individually to find the resulting components of the unit vector.  The resulting components should never be more than 1.

$$\hat{r} = \frac{\vec{r}}{|\vec{r}|} = \frac{<r_x, r_y, r_z>}{|\vec{r}|}$$