Difference between revisions of "Background Knowledge - The Basics of Vectors" - Rhea

Revision as of 15:24, 6 December 2010

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

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