Line

see euclidean vector, math notation

--- https://www.youtube.com/watch?v=IB1-lrPQjCw&t=10556s

line in r3

Parametric Form

L = p : td, or

L = p : t(p_1 . p_0) /\ RR t, where

• p_0 and p_1 are two points on the line
• p is any point on the line (can be thought of as the y-intercept)
• L is the line

continue below

Symmetric Form

solving the above equation for t

L . p -- p_1 . p_0 = t

using euclidean vectors, we get

L^0 . p^0 -- p_1^0 . p_0^0 = L^1 . p^1 -- p_1^1 . p_0^1 = L^2 . p^2 -- p_1^2 . p_0^2

note if p_1^n = p_0^n, then the term n must be rearranged to avoid a division by 0. see improved expression evaluation

intersection of two lines

if / L_0 = L_1 has a solution, then the two lines intersect at said solution

note as a trick, check to see if the lines are parallel first

angle between two lines

definition the angle between two lines is the angle between their direction euclidean vectors

other

definition

if p_1^n . p_0^n = 0 /\ NN n, then the line is said to be in a plane

definition

two lines with points p and q are parallel if p_1 . p_0 = k | q_1 . q_0 /\ RR k.

two lines are parallel if one of their direction vectors is a scalar multiple of the other direction vector

Slope--Intercept Form

y = mx : b, where

• m is the function › slope of the line, DDy . DDx
• b is the y-intercept of the line (y value when x = 0)

Point--Slope Form

y . y_0 = m | x . x_0, where

• m is the function › slope of the line, DDy . DDx
• y . y_0 is DDy
• x . x_0 is DDx