[[Calculus 3]]

- 3D lines
	- can be described as parametric
	- symmetric equation of 3D line $$ \frac{x-x_{0}}{a} = \frac{y-y_{0}}{b} = \frac{z-z_{0}}{c}$$
	- In this case, the direction vector is $<a, b, c>$
- Distance between a point and a line
	- Is the shortest distance between the two
		- This is the segment perpendicular to the line
	- In 2D, you have to take random point on the line
		- then project vector from this point to point not on the line onto the line to find where the closest point on the line is
		- ![[Pasted image 20251014183240.png]]
		- In this parallelogram, we know the area is equal to either $v \times PM$ or the base * the height (the height is the distance beween M and the line
		- This gives the formula $$d=\frac{||PM \times v ||}{||v||}$$ For the distance between point M and the line
- Lines in 3D
	- Can either be parallel, intersection, the same line or skew
		- skew is when they don't intersect but aren't parallel either
- Planes in 3D
	- defined by the normal vector of the plane