calc 3 exam 1

equation of a sphere

The equation of a sphere in three-dimensional space is given by the formula $(x-h)^2 + (y-k)^2 + (z-l)^2 = r^2$, where $(h, k, l)$ is the center of the sphere and $r$ is the radius.

unit vector equation

A unit vector equation represents a vector of length one in a specified direction, typically expressed in the form $\frac{v}{||{v}||}$

Dot product formula for angle between vectors

cos(theta) = a dot b/|a||b|

scalar projection of a onto b

a dot b/|b|

vector projection of a onto b

(a dot b)/(|b|²)*vector b

area of a parallelogram

|uXv|

vector eq of a line in 3d

r=r0 +tv

eq for a plane thru a point

a(x-x0)+b(y-y0)+c(z-z0)=0

T(t)

r’(t)/||r’(t)||

N(t)

T’(t)/||T’(t)||

B(t)

T(t) X N(t) OR (r’(t) X r’’(t))/||r’(t) X r’’(t)||

K(t)

||T’(t)||/||r’(t)||

arc length formula

L= Integral from a to b of ||r'(t)|| dt