Dot Product Applications (in R^2 and R^n)

In $2$-D, if two vectors are perpendicular with each other, i.e., the angle between them is $90^\circ$, we say they are orthogonal.

For example, consider the vectors 

$$\vec{v} = \begin{bmatrix} 1 \\ 2 \end{bmatrix} \quad \text{and} \quad \vec{w} = \begin{bmatrix} -2 \\ 1 \end{bmatrix}.$$

We have:

Since $\theta = 90^\circ$ between $\vec{v}$ and $\vec{w}$, notice from the dot product formula we have:

$$\vec{v} \cdot \vec{w} = \|\vec{v}\| \|\vec{w}\| \cos(90^\circ)$$

and so,

$$\vec{v} \cdot \vec{w} = 0$$

since $\cos(90^\circ) = 0$.

In higher dimensions, we actually use this $\vec{v} \cdot \vec{w} = 0$ condition to define two vectors being orthogonal. 

That is: for two $n$-dimensional vectors, 

$$...$$