| 719 |
vector between two points.
The vector
$\vect{v}=\overrightarrow{PQ}$
between two points
$P_1\left(x_1,y_1\right)$
and
$P_2\left(x_2,y_2\right)$
|
\[
v=\left(x_2-x_1,y_2-y_1\right)
\]
|
| 725 |
vector tangent to a curve at a point
$x=c$
|
\[
\vect{v}=\left(a,b\right)\\
\text{where}\\
y=f\left(x\right)\text{ and }
\left.\frac{dy}{dx}\right|_{x=c}=\frac{b}{a}
\]
|
| 725 |
vector normal to a vector
$\vect{u}$.
If $\vect{n}$ is a vector normal to $\vect{u}$
then so is $-\vect{n}$ and the implication holds.
|
\[
\vect{u}=\left(a,b\right)\Rightarrow\vect{n}
=\left(-b,a\right)\land-\vect{n}
=\left(b,-a\right)
\]
|
| 725 |
vector normal to a curve at a point
$P_0.$
If $\vect{v}=\left(a,b\right)$
is a vector tangent to a curve at
$x=c$ then $\vect{n}$ and $-\vect{n}$
are vectors normal to the curve at that point.
|
\[
\vect{n}=\left(-b,a\right)
\text{ and }
-\vect{n}=\left(b,-a\right)
\]
|
| 721 |
vector components from angle with
$x$-axis.
$\vect{v}$ forms an angle
$\theta$
with the
$x$-axis
(or polar axis)
|
\[
\vect{v}=
\left(
\abs{\vect{v}}
\cos\theta,
\abs{\vect{v}}
\sin{\theta}
\right)
\]
|
| 737 |
equation of line perpendicular to a vector.
The line and the vector are perpendicular.
|
\[
Ax+By=C\\
\vect{n}=A\vect{i}+B\vect{j}
\]
|
| 737 |
equation of line parallel to a vector.
The line and the vector are parallel.
|
\[
Bx-Ay=C\\
\vect{n}=A\vect{i}+B\vect{j}
\]
|
| 729 |
angle between vectors
|
\[
\theta=\cos^{-1}
\frac{\vect{u}\cdot\vect{v}}{\abs{\vect{u}}\abs{\vect{v}}}
\]
|
| 733 |
projection of $\vect{u}$ onto $\vect{v}$
|
\[
\text{proj}_\vect{v}\vect{u}
= \left(
\frac{\vect{u}\cdot\vect{v}}
{\abs{\vect{v}}^2}
\right)
\vect{v}
\]
|
| 733 |
scalar component of $\vect{u}$ in direction of $\vect{v}.$
If $\theta$ is the angle between $\vect{u}$ and $\vect{v}$
then the scalar component of $\vect{u}$ in the direction of
$\vect{v}$ is defined as given.
|
\[
\abs{\vect{u}}\cos\theta
=\frac{\vect{u}\cdot\vect{v}}{\abs{\vect{v}}}
\]
|
| 738 |
position vector
|
\[
\vect{r}\left(t\right)
=f\left(t\right)\vect{i}+g\left(t\right)\vect{j}
\]
|
| 743 |
velocity vector
|
\[
\vect{v}\left(t\right)
=\vect{r}'
=\frac{d\vect{r}}{dt}
=\abs{\vect{v}}\frac{\vect{v}}{\abs{\vect{v}}}
\]
|
| 743 |
speed
|
\[
\abs{\vect{v}\left(t\right)}
\]
|
| 743 |
acceleration vector
|
\[
\vect{a}\left(t\right)
=\vect{v}'\left(t\right)
=\frac{d\vect{v}}{dt}
=\vect{r}''
=\frac{d^2\vect{r}}{dt^2}
=\abs{\vect{a}}\frac{\vect{a}}{\abs{\vect{a}}}
\]
|
| 743 |
smooth.
If $\vect{r}'\left(t\right)=\zeros$
then $\vect{r}$ is not smooth.
|
\[
\]
|
| 743 |
unit vector in direction of motion
|
\[
\vect{T}=\frac{\vect{v}}{\abs{\vect{v}}}
\]
|
| 739 |
spiral of archimedes
|
\[
\vect{r}\left(t\right)
=\left(t\cos t\right)\vect{i}
+\left(t\sin t\right)\vect{j},
\quad
t\gt0
\]
|
|
initial vertical velocity
|
\[
v_{y_i}=v_0\sin\alpha
\]
|
|
initial horizontal velocity
|
\[
v_{x_i}=v_0\cos\alpha
\]
|
|
initial velocity
|
\[
\begin{align*}
\vect{v}_0
&=v_{x_i}\vect{i}+v_{y_i}\vect{j}\\
&=\left(v_0\sin\alpha\right)\vect{i}
+\left(v_0\cos\alpha\right)\vect{j}
\end{align*}
\]
|
|
position vector
(vector equation for ideal projectile motion) and displacement vector
|
\[
\Delta\vect{r}=\vect{r}-\vect{r}_0=\vect{v}_0+\frac{1}{2}\vect{g}t^2\\
\vect{r}=\vect{r}_0+\vect{v}_0+\frac{1}{2}\vect{g}t^2
\]
|
|
horizontal displacement
|
\[
\begin{align*}
x &=x_0+v_{x_i}t\\
\Delta x &=x-x_0\\
&=v_{x_i}t\\
&=\left(v_0\cos\alpha\right)t
\end{align*}
\]
|
|
vertical displacement
|
\[
\begin{align*}
y &=y_0+v_{y_i}t-\frac{1}{2}gt^2\\
\Delta y &=y-y_0\\
&=v_{y_i}t-\frac{1}{2}gt^2\\
&=\left(v_0\sin\alpha\right)t
-\frac{1}{2}gt^2
\end{align*}
\]
|
|
launch angle (firing angle, angle of elevation)
|
\[
\alpha=\tan^{-1}\frac{v_y}{v_x}
\]
|
|
time when maximum height is reached
|
\[
t_\text{halfway}=\frac{v_0\sin\alpha}{g}
\]
|
|
flight time
|
\[
t_f=2\cdot\frac{v_0\sin\alpha}{g}
\]
|
|
maximum height
|
\[
y_\text{max}=\frac{\left(v_0\sin\alpha\right)^2}{2g}
\]
|
|
range
|
\[
R=\frac{v_0^2}{g}\sin 2\alpha
\]
|
| 773 |
area between origin and a polar curve
|
\[
\]
|
| 775 |
area between two polar curves
|
\[
\]
|
Alex Notes