The Basics of Vectors
First, let
- $ \mathbf{v} = \begin{bmatrix} a \\ b \end{bmatrix} $
which denotes a vector between a point and the origin.
Then the length of this vector is given by
- $ \mathbf{\|v\|} = \sqrt{x^2 +y^2} $.
The same concept applies for two-point case between points v and w as shown below.
Let
- $ \mathbf{w} = \begin{bmatrix} c \\ d \end{bmatrix} $.
We have,
- $ \mathbf{v-w} = \begin{bmatrix} a-c \\ b-d \end{bmatrix} $,
and
- $ \mathbf{\|v-w\|} = \sqrt{(a-c)^2 +(b-d)^2} $.
In short, there is no significant difference in the three dimensional approach of the form
- $ \mathbf{v} = \begin{bmatrix} p \\ q \\ r \end{bmatrix} $.
Another essential concept is the angle between two vectors as shown in the following formula:
- $ \cos{\alpha} = \frac{\langle\mathbf{v}\, , \mathbf{w}\rangle}{\|\mathbf{v}\| \, \|\mathbf{w}\|} $.
This will be more prominent as we go through Inner Product Spaces section.
Lastly, a unit vector is a vector that has magnitude one and denoted as in the following:
- $ \boldsymbol{\hat{w}} = \frac{\boldsymbol{w}}{\|\boldsymbol{w}\|} $.
Ryan Jason Tedjasukmana
Back to Inner Product Spaces and Orthogonality
