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| Line 6: | Line 6: | ||
:<math>\mathbf{v} = \begin{bmatrix} | :<math>\mathbf{v} = \begin{bmatrix} | ||
a \\ | a \\ | ||
| − | b \end{bmatrix}</math> | + | b \end{bmatrix}</math> |
| − | + | ||
which denotes a vector between a point and the origin. | which denotes a vector between a point and the origin. | ||
| + | |||
Then the length of this vector is given by | Then the length of this vector is given by | ||
:<math>\mathbf{\|v\|} = \sqrt{x^2 +y^2}</math>. | :<math>\mathbf{\|v\|} = \sqrt{x^2 +y^2}</math>. | ||
| − | |||
| + | The same concept applies for two-point case between points '''v''' and '''w''' as shown below: | ||
Let | Let | ||
:<math>\mathbf{w} = \begin{bmatrix} | :<math>\mathbf{w} = \begin{bmatrix} | ||
c \\ | c \\ | ||
d \end{bmatrix}</math>. | d \end{bmatrix}</math>. | ||
| − | We have | + | We have |
:<math>\mathbf{v-w} = \begin{bmatrix} | :<math>\mathbf{v-w} = \begin{bmatrix} | ||
a-c \\ | a-c \\ | ||
| Line 25: | Line 25: | ||
and | and | ||
:<math>\mathbf{\|v-w\|} = \sqrt{(a-c)^2 +(b-d)^2}</math>. | :<math>\mathbf{\|v-w\|} = \sqrt{(a-c)^2 +(b-d)^2}</math>. | ||
| + | |||
In short, there is no significant difference in the three dimensional approach of the form | In short, there is no significant difference in the three dimensional approach of the form | ||
| Line 32: | Line 33: | ||
r \end{bmatrix}</math>. | r \end{bmatrix}</math>. | ||
| − | Another essential concept is the angle between two vectors as shown in the following formula: | + | |
| + | Another essential concept is the angle (e.g. α) between two vectors as shown in the following formula: | ||
:<math> \cos{\alpha} = \frac{\langle\mathbf{v}\, , \mathbf{w}\rangle}{\|\mathbf{v}\| \, \|\mathbf{w}\|}</math>. | :<math> \cos{\alpha} = \frac{\langle\mathbf{v}\, , \mathbf{w}\rangle}{\|\mathbf{v}\| \, \|\mathbf{w}\|}</math>. | ||
| + | This will be more prominent as we go through the Inner Product Spaces section. | ||
| − | |||
Lastly, a unit vector is a vector that has magnitude one and denoted as in the following: | Lastly, a unit vector is a vector that has magnitude one and denoted as in the following: | ||
:<math alt=>\boldsymbol{\hat{w}} = \frac{\boldsymbol{w}}{\|\boldsymbol{w}\|}</math>. | :<math alt=>\boldsymbol{\hat{w}} = \frac{\boldsymbol{w}}{\|\boldsymbol{w}\|}</math>. | ||
| + | |||
| + | |||
| + | '''Main Reference''' | ||
| + | ---- | ||
| + | Kolman, B., & Hill, D. (2007). ''Elementary linear algebra with applications (9th ed.)''. Prentice Hall. | ||
Latest revision as of 13:31, 8 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 (e.g. α) 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 the 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}\|} $.
Main Reference
Kolman, B., & Hill, D. (2007). Elementary linear algebra with applications (9th ed.). Prentice Hall.
Ryan Jason Tedjasukmana
Back to Inner Product Spaces and Orthogonal Complements
