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Arc length of a vector function <x(t), y(t), z(t)>

Unit tangent vector T of r(t)

Curvature given arc length parametrization and T

How to relate arc length to r(t)
ds/dt = r'(t)

Curvature given T and r(t)

Curvature given only r(t)

Curvature of a plane curve given y = f(x)


Find B given N and T
B = N x T

How to find the normal plane
Use T(t) as the normal and r(t) for a given point t on the curve

How to find the osculating plane
Use B(t) as the normal and r(t) for a given point t

v(t) given r(t)
v(t) = r'(t)

a(t) given r(t)
a(t) = r''(t) = v'(t)

v given T
V = vT = vT

Tangential component of acceleration given r(t) and T
r'(t) T = T

normal component of acceleration given r(t)


Equation of a sphere in R3 with radius r

How to compute the projection of a figure onto a given plane
Drop the nonincluded coordinate to 0


a dot (b+c)
a dot b + a dot c

c(a dot b)
c(a) dot b = a dot cb


Two vectors are orthogonal iff
a dot b = 0

How to find direction cosines
a<cos alpha, cos beta, cos gamma>

scalar projection of b onto a

vector projection of b onto a
(scalar proj * unit vector of a)

identity for the angles of direction cosines


Relating a x b to the angle
a x b = absin(theta)

Two nonzero vectors are parallel iff
a x b = 0

Area of a parallelogram determined by a and b
A = a x b

T/F the cross product is commutative
F

(ca) x b =
c(a x b) = a x (cb)


(a + b) x c =
a x c + a x b

a dot (b x c) =
(a x b) dot c

a x (b x c) =
(a dot c)b  (a dot b)c

Volume of a parallelepiped
Scalar triple product = a dot (b x c) (l, w, h)

Torque =
r x F = r F sin(theta)

Area of a triangle with vectors at a common vertex
area = a x b

Vectorparametric equation of a line
r(t) = r(0) + vt

Symmetric eqn of a line
 (x  x0)/a = (y  y0)/b = (z  z0)/c
 If a, b, or c=0, then the equality between the other two is preserved and we get something like x = x0, y(terms) = z(terms)

Vectorparametric equation of a line segment
r(t) = (1t)r0 + tr1

How to verify skew lines
First, ensure that they never intersect (in space, not necessarily in time). Then ensure that they aren't parallel

Vector equation of a plane
n dot r = n dot r0

scalar eqn of plane
 a(xx0) + b(yy0) + c(zz0) = 0
 n = <a,b,c>

linear eqn of a plane
ax + by + cz + d = 0

How to determine parallelity of planes
if their normal vectors are parallel

How to determine angle between 2 planes
use the normal vectors and the angledotproduct formula

How to find the line of intersection of 2 planes
Solve for one of the variables (x,y,z) in terms of the others and use it as a parameter

Distance from a point to a plane
(or between 2 parallel planes)

Distance between skew lines
They define 2 planes. You just need a vector orthogonal to both skew lines, so cross them! That's n

