Calc 3 Midterm 1

symmetric line equation

(x-x0)/a = (y-y0)/b =. (z-z0)/c

Distance from a point to a line

(|r-r0 * v|)/(|v|)

Distance from a point to a plane

(|r-r0 * n|)/(|n))

Equation of line that represents the intersection between two vectors

find n1 and n2, then cross them to find v and use an arbitrary point to construct a line

Equation of a plane that contains 3 points

Common Knowledge

Equation of a plane that contains two parallel lines

take two arbitrary points from the lines r1 r2, cross them with one of the lines direction vector, that gives you n

Equation of a plane that contains line and point=r0

choose r1 on line, r1-r0 * v = n

i magnitude of r(t) is constant

r(t) cross v(t) is zero

Smoothness condition

Curve is smooth if magnitude of dr/dt is nonzero

Distance between two lines

choose two random points on the line and use a direction vector (|v * (r1-r2)|)/(|v|))

Arclength

integral of the magnitude of the position vector with respect to the parameter

dr/ds is equivalent to

the velocity divided by the magnitude of the velocity

dT/ds is equivalent to

second derivative of the position vector with respect to arclength, kappa times N, curvature times normal vector

kappa and normal defintions

kappa is the magnitude of the derivative of the tangent vector with respect to arclength, normal is the direction

Different variations of B = T cross N

T = N cross B, N = B cross T

kappa

(|v* a|)/(|v|³)

B (Binormal Vector)

(v*a)/(|v*a|)

N (Normal Vector) V=speed

(Va -vV’)/(kr³)

curvature defintion in function form

(|f’’’(t))/(sqrt(1+(f(t))²)³