1.1 Vectors

class: center, middle

# Engineering Physics - Fall 2024

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## Dynamics:
Motion of bodies under the action of forces.

### Vectors:
Mathematical representations of the state of a body.

How would **You** quantify the position and <a style="color:tomato;">velocity</a> of a point in this room?
 
 `$\vec{r}=(r_x,r_y,r_z)$` where the `$r_i$`'s are the <a style="color:tomato;">components</a> of the vector `$\vec{r}$`
 in the direction of the coordinate axes `$\hat{x}$`, `$\hat{y}$`, and `$\hat{z}$`.

.center[[The same point `$P$` represented in two coordinate systems.](/images/vec-1.svgz)]

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#### Addition/Subtraction:

What is the "net" effect of various pulls on the same point? 
 In general, how do vectors combine?

`$\vec{a}+(\vec{b}+\vec{c})=(\vec{a}+\vec{b})+\vec{c}=\begin{pmatrix} a_x+b_x+c_x \\ x\leftrightarrow y \\ x\leftrightarrow z \end{pmatrix}$`
 
<hr style="margin-top: 10px;margin-bottom: 60px;" />

.center[![Center-aligned image](/images/vec-add.svgz)]

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#### Scalar/dot product:
 

What is the "net" overlap between two vectors? 
Prosaically, how alike are 2 vectors?

.center[$$\vec{A}\cdot\vec{B}=A_xB_x+A_yB_y+A_ZB_z=\vert\vec{A}\vert\,\vert\vec{B}\vert\cos\theta$$]

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#### Cross product:
 

How large of a space do the 2 vectors span?
Prosaically, how much volume can be addressed by combinations

.center[$$c_1\vec{a}+c_2\vec{b}$$]

with $-1\leq c_{1,2}\leq 1$?

$\vec{a}\times\vec{b}=\vec{n}_{\tiny ab}\,\vert\vec{a}\vert\,\vert\vec{b}\vert\,\vert\sin\theta\vert=
\begin{pmatrix}a_yb_z-b_ya_z\\a_zb_x-b_za_x\\a_xb_y-b_ya_x\end{pmatrix}~\text{in 3D}$

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#### Derivative of a vector:
 

How do You quantify <a style="color:tomato;">infinitesimal </a> changes of a vector?


Change   $\Delta\vec{r}=\vec{r}_2-\vec{r}_1$   per increment (time)   $\Delta t=t_2-t_1$   is an average!

Velocity:   `$\vec{v}=\lim\limits_{\Delta t\to0}\frac{\Delta\vec{r}}{\Delta t}=
\lim\limits_{\Delta t\to0}\begin{pmatrix}\frac{r_{2,x}-r_{1,x}}{t_2-t_1}\\\frac{r_{2,y}-r_{1,y}}{t_2-t_1}\end{pmatrix}=\frac{d\vec{r}}{dt}=\begin{pmatrix}\frac{dr_x}{dt}\\\frac{dr_y}{dt}\end{pmatrix}$`

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### Ponderables:

1. Select 5 criteria to characterize your fellow student (e.g., height, hair colour, handedness, hobby, ...) and define
a data vector

$\vec{s}=\begin{pmatrix}\text{criterion}~1\\\text{criterion}~2\\\vdots\end{pmatrix}$

- Define <a style="color:tomato;">addition and multiplication</a> for your criteria.
 - When do two individuals have the <a style="color:tomato;">least in common</a>?
 - When has the combination of two individuals the <a style="color:tomato;">highest potential</a> to address something new?

2. The [set](/data/motion.dat) represents the coordinates of an object at successive time instances.

- <a style="color:tomato;">Draw</a> those number pairs as vectors <a style="font-size:12pt;color:gray;">(left/right column `$\to~x/y$` coordinate)</a>.
 - What kind of shape does the motion remind you of and how would you represent this in the form: <a style="color:tomato;">`$r_y=f(r_x)$`</a>?
 - Is the velocity and/or speed of this motion <a style="color:tomato;">constant</a>?