# Calc III - Ch13 formulae.txt

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1. Arc length of a vector function <x(t), y(t), z(t)>
2. Unit tangent vector T of r(t)
3. Curvature given arc length parametrization and T
4. How to relate arc length to r(t)
ds/dt = ||r'(t)||
5. Curvature given T and r(t)
6. Curvature given only r(t)
7. Curvature of a plane curve given y = f(x)
8. Find given only r(t)
9. Find B given N and T
B = N x T
10. How to find the normal plane
Use T(t) as the normal and r(t) for a given point t on the curve
11. How to find the osculating plane
Use B(t) as the normal and r(t) for a given point t
12. v(t) given r(t)
v(t) = r'(t)
13. a(t) given r(t)
a(t) = r''(t) = v'(t)
14. v given T
V = vT = ||v||T
15. Tangential component of acceleration given r(t) and T
||r'(t)||T = T
16. normal component of acceleration given r(t)
17. Distance formula in R3
18. Equation of a sphere in R3 with radius r
19. How to compute the projection of a figure onto a given plane
Drop the non-included coordinate to 0
20. A dot A
||A||^2
21. a dot (b+c)
a dot b + a dot c
22. c(a dot b)
c(a) dot b = a dot cb
23. Angle between vectors
24. Two vectors are orthogonal iff
a dot b = 0
25. How to find direction cosines
||a||<cos alpha, cos beta, cos gamma>
26. scalar projection of b onto a
27. vector projection of b onto a
(scalar proj * unit vector of a)
28. identity for the angles of direction cosines
29. a x a
0-vector
30. Relating a x b to the angle
a x b = ||a||||b||sin(theta)
31. Two nonzero vectors are parallel iff
a x b = 0
32. Area of a parallelogram determined by a and b
A = a x b
33. T/F the cross product is commutative
F
34. (ca) x b =
c(a x b) = a x (cb)
35. a x (b+c)
a x b + a x c
36. (a + b) x c =
a x c + a x b
37. a dot (b x c) =
(a x b) dot c
38. a x (b x c) =
(a dot c)b - (a dot b)c
39. Volume of a parallelepiped
Scalar triple product = |a dot (b x c)| (l, w, h)
40. Torque =
r x F = ||r|| ||F|| sin(theta)
41. Area of a triangle with vectors at a common vertex
area = |a x b|
42. Vector-parametric equation of a line
r(t) = r(0) + vt
43. 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)
44. Vector--parametric equation of a line segment
r(t) = (1-t)r0 + tr1
45. How to verify skew lines
First, ensure that they never intersect (in space, not necessarily in time). Then ensure that they aren't parallel
46. Vector equation of a plane
n dot r = n dot r0
47. scalar eqn of plane
• a(x-x0) + b(y-y0) + c(z-z0) = 0
• n = <a,b,c>
48. linear eqn of a plane
ax + by + cz + d = 0
49. How to determine parallelity of planes
if their normal vectors are parallel
50. How to determine angle between 2 planes
use the normal vectors and the angle-dot-product formula
51. 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
52. Distance from a point to a plane
(or between 2 parallel planes)
53. Distance between skew lines
They define 2 planes. You just need a vector orthogonal to both skew lines, so cross them! That's n
 Author: broach13 ID: 235892 Card Set: Calc III - Ch13 formulae.txt Updated: 2013-09-22 21:03:12 Tags: calc iii formula Folders: Description: formula for calc iii ch 13 Show Answers: