Eqtns to memorize for lines+planes in space

Distance from point to line

\frac{\left\vert PScrossv\right\vert}{\left\vert v\right\vert}

Plane eqtn with 3 points

Point 1: Center
Point 1 —> Point 2: Find vector form

Point 2 —> Point 3: Find vector form

Find cross product of both vectors to find n —> component perpendicular

Plane eqtn with point and line in it

Point: Center
Line: r0 + vt form —> v given by constant multiplying t for each x, y, z. The line is parallel to the plane —> we can use this eqtn to find another line parallel w/the point (set line to 0, then use point). Take cross product to find n perpendicular to plane.

Plane eqtn with two lines in it

Find vectors for both lines —> cross product to get vector perpendicular to entire plane. This is n. Use point in one line as center

Distance b/w plane and a point

\frac{\left\vert SP\cdot n\right\vert}{\left\vert n\right\vert} SP is the vector from any point on the plane to the point off the plan, n is perpendicular normal vector.

Parallel planes

Same normal vectors —> can check w formatting

Intersections for planes

If we know they aren’t parallel — add planes 

Ideal projectile motion

r = ( x0 + v0 cos a)ti + (y0 + (vo sina)t - (gt²)/2)j

u cross (v cross w) =

(u dot w) v - (u dot v) w

area of triangle

\frac{\left|ucrossv\right|}{2}

Torque

r cross F

box product (volume of parallelpoid)

|(u cross v) dot w|

Work

Force dot distance

max height

(v0 sin a)²/2g

Flight time

t= (2v0 sin a)/g

Range

(v0² sin 2a)/g