MVC Test 2

What do you need to find the equation of a line?

A point and a parallel vector

Vector Equation

<x,y,z> = <x_o,y_o,z_o> + t <a,b,c>

Parametric Equations

x=x_o+at

y=y_o+bt

z=z_o+ct

Symmetric Equations

\frac{x-xo}{a}=\frac{y-yo}{b}=\frac{z-zo}{c}

How to find a parallel vector passing through points A & B?

Vector = AB

How to find where a line intersects xy plane?

Set z=0 in symmetric equation then solve for x & y

What do you need to find the equation of a plane?

A point and a perpendicular vector 

Normal Vector

Perpendicular (orthogonal) vector

Dot product for perpendicular vectors

=0

Equation of a plane

a(x-x_o) + b(y-y_o) + c(z-z_o) = 0

Points P, Q, & R are co-linear if?

PQ = k*PR

Linear Equation of a plane

ax+by+cz=d

If points are co-linear what does this mean for the plane?

It is not a plane, it is a line 

What is the normal vector of a plane that passes through points P, Q, R?

PR x PQ

Planes are parallel when?

Their normal vectors are parallel

How to find the line of intersection of 2 planes?

Set z=0 to find the point (P) that lies on both planes. P is perpendicular to both normal vectors.

Then use the cross product of the normal vectors (n1 x n2) to get the parallel vector (v)

Use P and v to get the eq. of the line

Distance b/w point and plane

\frac{\overrightarrow{n}\cdot\overrightarrow{QP}}{\left\vert\overrightarrow{n}\right\vert}=\frac{\left\vert ax+by+cz+d\right\vert}{\sqrt{a^2+b^2+c^2}}

Unit Tangent Vector

\overrightarrow{T}\left(t\right)=\frac{\overrightarrow{r^{\prime}}}{\left\vert\overrightarrow{r}^{\prime}\right\vert}

Differentiation rules of dot product

\overrightarrow{u}\cdot\overrightarrow{v^{\prime}}+\overrightarrow{u^{\prime}}\cdot\overrightarrow{v}

Differentiation rules of cross product

\overrightarrow{u}\chi\overrightarrow{v^{\prime}}+\overrightarrow{u^{\prime}}\chi\overrightarrow{v} 

can’t change order 

What are the 4 derivative forms of  \overrightarrow{r}

r  - position

r^{\prime}  - velocity 

\left\vert r^{\prime}\right\vert  - speed

r^{\prime\prime}  - acceleration

Speed of gravity

9.8 m/s²

Vertical position as a function of time and variable definitions

y\left(t\right)=y_{o}+V_{oy}t-\frac12gt^2

y_o= Initial height

V_oy= Vertical y velocity

g = gravity

Solving for time of flight

y(t) = 0

Range of flight equation & variables

R = V_ox* t

Range = horizontal x velocity * time of flight

Max height equation, variables, & how to solve

V_y(t) = V_oy - gt_peak

Max height = Vertical y velocity - gravity (time @ peak)

Set to 0 to solve for t_peak

Solve for y(t_peak) to solve for height

Speed of impact equation

< V_ox , V_oy - gt >

< horizontal velocity , vertical velocity - gravity (time of impact) >

Magnitude = speed

Position of flight equation

< V_ox , V_oy - gt >

< horizontal velocity , vertical velocity - gravity (time of flight) >

Horizontal position as a function of time and variable definitions

x(t) = V_ox * t

Velocity of x * time

Arc Length Formula

L=\int\left\vert\overrightarrow{r^{\prime}}\right\vert from a to b

total distance along a curve

s\left(t\right)=\int\overrightarrow{\left\vert r^{\prime}\left(u\right)\right\vert} from a to t

distance so far along a curve

\overrightarrow{N} (normal vector)

\overrightarrow{N}=\frac{\overrightarrow{T}}{\left\vert\overrightarrow{T}\right\vert}

\overrightarrow{B} Binormal Vector

\overrightarrow{B}=\overrightarrow{v}\chi\overrightarrow{a} (easier)

\overrightarrow{B}=\overrightarrow{T}\chi\overrightarrow{N}

Normal Plane vector

= T

Osculating Plane vector

= B

Osculating Circle

r=\frac{1}{\kappa}

circle with radius r that best fits the curve @ a location

Quadratic Formula

\frac{-b+\sqrt{b^2-4ac}}{2a}

Quotient Rule y=\frac{u}{v}

\frac{vu^{\prime}-uv^{\prime}}{v^2}

\frac{bottom\cdot top^{\prime}-top\cdot bottom^{\prime}}{bottom^2}

Equation for moving units along a curve

s\left(t\right)=\int\left\vert\overrightarrow{r\left(u\right)}^{\prime}\right\vert  from a to t

must consider what t value allows for the initial position this = a

At inflection point

y’’ = 0

Range of ln

\left(-\infty,\ln\left(u_{\max}\right)\right) 

where u_max is the domain parameter

Linearization equation & what is it equivalent to?

L=f\left(x_{o},y_{o}\right)+f_{x}\left(x_{o},y_{o}\right)\left(x-x_{o}\right)+f_{y}\left(x_{o},y_{o}\right)\left(y-y_{o}\right) Tangent Plane

Measuring error

dz=\frac{\delta z}{\delta x}dx+\frac{\delta z}{\delta y}dy

Volume of a box

V=lwh

Surface Area of a box

= 2lh + 2lw +2hw

Volume of a closed cylinder

V=\pi r^2h

Space Diagonal

D=\sqrt{l^2+w^2+h^2}