Vectors 2:

Created by: Catchupman
Created on: 08-10-17 13:17

Nat 5 info:

A vector is a quantity which has both direction and magnitude.

The magnitude of a vector is its size or length.
A directed line segment from A to B is defined as $LaTeX: $\overrightarrow {AB} $$ .
A vector or force can also be defined by a lower-case letter in bold.
Vectors are equal if they have the same direction and magnitude.
When a vector has its direction reversed it is negative e.g. a becomes -a.
A vector can be multiplied by a scalar e.g. doubling a gives 2a.
Vectors can be added by joining one to the end of another.
Displacement is the shortest distance from A to B.
A vector journey is a description of its displacement.
The components of a vector describe the journey from A to B e.g. $LaTeX: $\left( {\begin{array}{{20}{c}} x \\ y \end{array}} \right)$$ in 2D or $LaTeX: $\left( {\begin{array}{{20}{c}} x \\ y \\ z \end{array}} \right)$$ in 3D.
Arithmetic can be performed on the components $LaTeX: $\left( { + \; - \; \times \; \div } \right)$$ e.g.
$LaTeX: $\left( {\begin{array}{{20}{c}} 1 \\ 2 \end{array}} \right)\,\, + \,\,\left( {\begin{array}{{20}{c}} 4 \\ 3 \end{array}} \right)\;\; = \;\;\left( {\begin{array}{{20}{c}} {1\,\, + \,\,4} \\ {2\,\, + \,\,3} \end{array}} \right)\;\; = \;\;\left( {\begin{array}{{20}{c}} 5 \\ 5 \end{array}} \right)$$
The magnitude is calculated from the components using the Theorem of Pythagoras e.g. $LaTeX: ${\mathbf{u}}\;\; = \;\;\left( {\begin{array}{{20}{c}} x \\ y \end{array}} \right)$$ , $LaTeX: $\left| {\mathbf{u}} \right|\;\; = \;\;\sqrt {{x^2}\,\, + \,\,{y^2}} $$ and $LaTeX: ${\mathbf{v}}\;\; = \;\;\left( {\begin{array}{{20}{c}} x \\ y \\ z \end{array}} \right)$$ , $LaTeX: $\left| {\mathbf{v}} \right|\;\; = \;\;\sqrt {{x^2}\,\, + \,\,{y^2}\,\, + \,\,{z^2}} $$