Formula Summary

Table of Content

Points

Mid Point

M_{P Q} = \frac{P + Q}{2} = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})

Distance

d (P, Q) = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}

Vectors in 2D and 3D

Length

∥ \vec{u} ∥ = \sqrt{x^{2} + y^{2}}

Dot product (Info)

\vec{a} \cdot \vec{b} = a_{x} b_{x} + a_{y} b_{y}

Result (Scalar)	Angle between $\vec{a}$ and $\vec{b}$ is...
$> 0$	$< 90^{\circ}$
$= 0$	$= 90^{\circ}$
$< 0$	$> 90^{\circ}$

Angle between 2 vectors

θ = \cos^{- 1} (\frac{\vec{a} \cdot \vec{b}}{∥ \vec{a} ∥ ∥ \vec{b} ∥})

Special cases

Dot product result	Angle
-1	$180^{\circ}$
0	$90^{\circ}$
1	$0^{\circ}$

^ ONLY WORKS FOR UNIT VECTORS ^

For dot product 1 and -1 NEED to be unit vectors. For dot product 0, it doesn't matter

Checking if vectors are parallel/perpendicular

For parallel

$\vec{u} = c \vec{v}$
$\vec{u}$ must be a scalar of $\vec{v}$

\vec{u}

and

\vec{v}

are unit vectors, can use Dot product, Special cases

For perpendicular

$\vec{u} \cdot \vec{v} = 0$
See Dot product, Special cases

Vector Projection (Info)

Project $\vec{b}$ onto $\vec{a}$ ,

General:

p r o j_{\vec{a}} \vec{b} = \frac{\vec{b} \cdot \vec{a}}{∥ \vec{a} ∥^{2}} \vec{a}

If $\vec{a}$ is a unit vector:

p r o j_{\vec{a}} \vec{b} = (\vec{b} \cdot \hat{a}) \hat{a}

Checking if moving towards a point

Only 1 point moving

Get vector $\vec{Q P}$ ,
Find $\vec{Q P} \cdot \vec{v}$
At $Q$ , Result < 0, angle is < 90 $^{\circ}$ so moving towards
At $Q_{1}$ , Result = 0, angle is = 90 $^{\circ}$ . Neither moving towards/away. This is will only happen for a split second
At $Q_{2}$ , Result > 0, angle is > 90 $^{\circ}$ so moving away

2 points moving

Assume P is stationary **relative** to Q, Then same as [[#Only 1 point moving]] except use:

Result = $\vec{v_{1}} - \vec{v_{2}} \cdot \vec{P Q}$

Result < 0, moving towards
Result = 0, Neither moving towards/away
Result > 0, moving away

What it means by relative

Imagine P and Q are cars at a intersection. And you are sitting in car P.
From my perspective, Car Q moves towards me relative to my (Car P) current speed.

Vectors in 2D

Making a normal vector

Swap $x$ and $y$
Negate either the $x$ or $y$
- Negate $x$ to rotate $90^{\circ}$ anti-clockwise
- Negate $y$ to rotate $90^{\circ}$ clockwise

Example

$\vec{d} = [\begin{matrix} 2 \\ 3 \end{matrix}] ⟹ \vec{n} = [\begin{matrix} 3 \\ - 2 \end{matrix}]$

Linear equation

For reference, this is the normal way to represent an equation

y = m x + c

Where

$m$ = slope and
$c$ = y-intercept

Find using $\vec{n}$ and 1 point (Proof)

\vec{x} \cdot \vec{n} = \vec{O P} \cdot \vec{n}

Expanded out:

a x + b y = a x_{0} + b y_{0}

Vector equation

Notation

\vec{x} = P + t \vec{d} O R \vec{x} = (x_{0}, y_{0}) + [\begin{matrix} a \\ b \end{matrix}]

Find vector equation using 2 points

\vec{x} = P + \vec{P Q}

Linear $\to$ Vector equation

\begin{aligned} a x + b y & = c \\ \vec{n} & = [\begin{array}{c} a \\ b \end{array}] \\ \vec{d} & = [\begin{array}{c} - b \\ a \end{array}] \\ Sub x=0 OR y=0 to & find P, \\ \vec{x} & = P + t \vec{d} \end{aligned}

Parametric equation

{\begin{cases} x = x_{0} + t a \\ y = y_{0} + t b \end{cases} t \in (- \infty, \infty)

Finding shortest distance

Point to line

Given $\vec{x} = P + t \vec{d}$ and $Q (x, y)$

Dist from Q to line = | | p r o j_{\vec{n}} \vec{P Q} | |

\vec{P Q}

direction

P is a point on the line as the normal starts from the line too

Checking if a moving point will hit line

Hit	Miss

Hit if the signs of these 2 are different

$\vec{Q P} \cdot \vec{n}$
$\vec{v} \cdot \vec{n}$
Or in code:

\vec{Q P} \cdot \vec{n} < 0! = \vec{v} \cdot \vec{n} < 0

Excalidraw - Full table

Vectors in 3D

Line

\vec{x} = P + t \vec{d}

No linear equation for line.

Linear equation in 3D is a Plane

Quick normal vector in 3D

Lines in 3D has infinite number of normal vectors

Steps (Similar to 2D):

Set one of the coords to 0,
Swap the other 2 coords
Negate one of them

Example

$\vec{d} = [\begin{matrix} 2 \\ 3 \\ 4 \end{matrix}] \to \vec{n} = [\begin{matrix} 0 \\ - 4 \\ 3 \end{matrix}]$

Plane

Notation

A plane equation can be represented by $α, β, γ$ , etc...

Vector equation

Using 2 direction vectors, $$\vec{x}=P+s\vec{u}+t\vec{v}$$ > [!WARNING] $\vec{u}$ and $\vec{v}$ cannot be parallel ##### 3 Points $\to$ Vector equation Given $P, Q, R$, $$ \vec{x}=P+s\vec{PQ}+t\vec{PR} $$

Vector $\to$ Linear equation

Find $\vec{n}$ using $\vec{u} \times \vec{v}$ and then form the Linear equation

\begin{aligned} \vec{n} & = \vec{u} \times \vec{v} \\ \vec{x} \cdot \vec{n} & = \vec{O P} \cdot \vec{n} \end{aligned}

Linear equation

Same as 2D, $$ \vec{x}\cdot \vec{n}=\vec{OP}\cdot \vec{n} $$ ### Cross product Used to find normal of 2 vectors

\vec{u} \times \vec{v} = [\begin{matrix} u_{x} \\ u_{y} \\ u_{z} \end{matrix}] \times [\begin{matrix} v_{x} \\ v_{y} \\ v_{z} \end{matrix}] = [\begin{matrix} u_{y} v_{z} - u_{z} v_{y} \\ u_{z} v_{x} - u_{x} v_{z} \\ u_{x} v_{y} - u_{y} v_{x} \end{matrix}]

To check, make sure

(\vec{u} \times \vec{v}) \cdot \vec{u} = 0

To visualize using right hand rule

4 fingers pointing the direction from $\vec{u} \to \vec{v}$
thumb points to the result of the cross product.

NOTE: $\vec{u} \times \vec{v} = - \vec{v} \times \vec{u}$

Related: Point to Line dist in 3D

Distances for lines

Point to Line

!03-L Week 3, p.47

--- Using cross product (recommended)
$d(P,l)=\frac{\text{Area of parallelogram}}{||\vec{d}||}=\frac{||\vec{QP}\times \vec{d}||}{||\vec{d}||}$

--- Using projection
$
\begin{align}
\vec{QH}&=proj_{\vec{d}}{\vec{QP}} \\
\vec{HP}&=\vec{HQ}+\vec{QP} \\
d(P,l)&=||\vec{HP}||
\end{align}
$

Line to Plane

See Plane to Line below

Line to Line

Given $l_{1} : \vec{x} = P + t \vec{d_{1}}$ and $l_{2} : \vec{x} = Q + t \vec{d_{2}}$ ,

If $d_{1}$ parallel to $d_{2}$
- Distance = $d (P, l_{2})$ - Point to Line
If $d_{1}$ NOT parallel to $d_{2}$ ,
- Make a plane containing $l_{2}$ that's parallel to $l_{1}$
  - $\vec{n_{α}} = \vec{d_{1}} \times \vec{d_{2}}$
  - $α : \vec{x} \cdot \vec{n_{α}} = Q \cdot \vec{n_{α}}$
- Distance = $d (P, α) = | | p r o j_{\vec{n_{α}}} \vec{Q P} | |$ - Plane to Point

Distances for planes

Plane to Point

Same as Point to line dist in 2D

Given $\vec{x} = P + t \vec{d}$ and $Q (x, y, z)$

Dist from Q to line = | | p r o j_{\vec{n}} \vec{P Q} | |

\vec{P Q}

direction

P is a point on the plane as the normal starts from the line too

Plane to Line, Plane to Plane

There are 2 cases:

If plane/line is NOT parallel to the other plane, distance = 0
If plane/line is parallel to the other plane,
- Treat it as Plane to Point $\to$ Distance = $| | p r o j_{{\vec{n}}_{β}} \vec{Q P} | |$

Examples:

--- Plane to Plane
Given planes 
- $\alpha:P,\vec{n}_{\alpha} \implies\vec{x}_{\alpha}\cdot \vec{n}_{\alpha}=\vec{x}_{\alpha}\cdot \vec{OP}$ 
- $\beta:Q,\vec{n}_{\beta}\implies\vec{x}_{\beta}\cdot \vec{n}_{\beta}=\vec{x}_{\beta}\cdot \vec{OQ}$

There are 2 cases:
- $\alpha$ and $\beta$ intersect $\to$ $\vec{n}_{\alpha}$ NOT parallel to $\vec{n}_{\beta}$
$
\text{So, }d(\alpha,\beta)=0
$
- $\alpha$ and $\beta$ are parallel $\to$ $\vec{n}_{\alpha}$ parallel to $\vec{n}_{\beta}$ 
$
d(\alpha,\beta)=d(P,\beta)=||proj_{\vec{n}_{\beta}}{\vec{QP}}||
$
This is same as Plane to Point (Treat the other plane as a point as it's parallel).
--- Plane to Line (css-class: pdf-page-break)
Given:
- $\alpha:P,\vec{n}_{\alpha} \implies\vec{x}_{\alpha}\cdot \vec{n}_{\alpha}=\vec{x}_{\alpha}\cdot \vec{OP}$ 
- $l:Q,\vec{d}\implies \vec{x}=Q+t\vec{d}$ 

There are 2 cases:
- $\alpha$ and $l$ intersect $\to$ $\vec{d}$ NOT $\perp$ $\vec{n}_{\alpha}$ 
$
\text{So, }d(l,\alpha)=0
$
- $\alpha$ and $l$ are parallel $\to$  $\vec{d} \perp \vec{n}_{\alpha}$ 
$
d(l,\alpha)=d(P,\alpha)=||proj_{\vec{n}_{\alpha}}{\vec{QP}}||
$

This is same as Plane to Point (Treat the other line as a point as it's parallel).

Others

Are 2 lines the same

$\vec{x} = P + s \vec{d_{1}}$ and $\vec{x} = Q + t \vec{d_{2}}$
Same if (both true):

$d_{1} | | d_{2}$
$P$ is a point on $\vec{x} = Q + t \vec{d_{2}}$

Table of Content

Points

Mid Point

Distance

Vectors in 2D and 3D

Length

Dot product (Info)

Angle between 2 vectors

Special cases

Checking if vectors are parallel/perpendicular

For parallel

For perpendicular

Vector Projection (Info)

Checking if moving towards a point

Only 1 point moving

2 points moving

Vectors in 2D

Making a normal vector

Linear equation

Find using n→ and 1 point (Proof)

Vector equation

Notation

Find vector equation using 2 points

Linear → Vector equation

Parametric equation

Finding shortest distance

Point to line

Checking if a moving point will hit line

Vectors in 3D

Line

Quick normal vector in 3D

Plane

Notation

Vector equation

Vector → Linear equation

Linear equation

Distances for lines

Point to Line

Line to Plane

Line to Line

Distances for planes

Plane to Point

Plane to Line, Plane to Plane

Others

Are 2 lines the same

Find using $\vec{n}$ and 1 point (Proof)

Linear $\to$ Vector equation

Vector $\to$ Linear equation