Question Number 5859 by wanderer last updated on 02/Jun/16
$${what}\:{is}\:{the}\:{intutive}\:{understanding}\:{of}\: \\ $$$${eigenvalues}\:{and}\:{vectors}.{with}\:{practical}\: \\ $$$${examples}. \\ $$
Commented by FilupSmith last updated on 02/Jun/16
![A vector is a direction and magnitude. Lets say you face North. if a= [(1),(0) ], a∈R^2 this can be interperated as 1 unit to the right. Application in Physics include, for example projectile motion, velocity, forces, etc. For example: F=ma F and a are vectors. So therefore the Force has the same direction as a but a different value/magnitued due to being multiplied by m. e.g. a=4 m/s^2 downwards m=2kg ∴ F=(4×2) Newtons downwards F=8 N downwards](Q5860.png)
$$\mathrm{A}\:\mathrm{vector}\:\mathrm{is}\:\mathrm{a}\:\mathrm{direction}\:\mathrm{and}\:\mathrm{magnitude}. \\ $$$$\mathrm{Lets}\:\mathrm{say}\:\mathrm{you}\:\mathrm{face}\:\mathrm{North}.\: \\ $$$$\mathrm{if}\:\:\:\:\:\:\boldsymbol{{a}}=\begin{bmatrix}{\mathrm{1}}\\{\mathrm{0}}\end{bmatrix},\:\:\:\:\boldsymbol{{a}}\in\mathbb{R}^{\mathrm{2}} \\ $$$$\mathrm{this}\:\mathrm{can}\:\mathrm{be}\:\mathrm{interperated}\:\mathrm{as}\:\mathrm{1}\:\mathrm{unit}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{right}. \\ $$$$ \\ $$$$\mathrm{Application}\:\mathrm{in}\:\mathrm{Physics}\:\mathrm{include},\:\mathrm{for}\:\mathrm{example} \\ $$$$\mathrm{projectile}\:\mathrm{motion},\:\mathrm{velocity},\:\mathrm{forces},\:\mathrm{etc}. \\ $$$$ \\ $$$$\mathrm{For}\:\mathrm{example}: \\ $$$$\boldsymbol{{F}}={m}\boldsymbol{{a}} \\ $$$$\boldsymbol{{F}}\:\:\:\mathrm{and}\:\:\:\boldsymbol{{a}}\:\mathrm{are}\:\mathrm{vectors}. \\ $$$$\mathrm{So}\:\mathrm{therefore}\:\mathrm{the}\:\mathrm{Force}\:\mathrm{has}\:\mathrm{the}\:\mathrm{same}\: \\ $$$$\mathrm{direction}\:\mathrm{as}\:\boldsymbol{{a}}\:\mathrm{but}\:\mathrm{a}\:\mathrm{different}\:\mathrm{value}/\mathrm{magnitued} \\ $$$$\mathrm{due}\:\mathrm{to}\:\mathrm{being}\:\mathrm{multiplied}\:\mathrm{by}\:{m}. \\ $$$${e}.{g}. \\ $$$$\boldsymbol{{a}}=\mathrm{4}\:\:\mathrm{m}/\mathrm{s}^{\mathrm{2}} \:\:\mathrm{downwards} \\ $$$${m}=\mathrm{2kg} \\ $$$$\therefore\:\boldsymbol{{F}}=\left(\mathrm{4}×\mathrm{2}\right)\:\mathrm{Newtons}\:\mathrm{downwards} \\ $$$$\boldsymbol{{F}}=\mathrm{8}\:\mathrm{N}\:\mathrm{downwards} \\ $$