Vectors Flashcards
Spherical polar coordinates
x=rsin(theta)cos(phi)
y=rsin(theta)sin(phi)
z=rcos(theta)
Cylindrical polar coordinates
x=rcos(theta)
y=rsin(theta)
z=z
a x (b x c)
(a . c)b - (a . b)c
(a x b) x c
(a . c)b - (b.c)a
Scalar triple product
a . b x c
Scalar product
a . b = mag(a)mag(b)cos(theta)
Vector product
a x b = mag(a)mag(b)sin(theta)
Equation of a line
r = a + k(b-a)
[get on to line and then move along]
Component of a in n̂
a.n̂
Projection of a in n̂
(a.n̂)n̂
[a.n̂ gives number - multiply by vector to give projection]
Equation of plane (point and normal)
r.n̂=a.n̂
Perpendicular distance to origin
r.n̂=a.n̂=p
Equation of a plane (3 points)
r-a=mu(b-a)+ upsilon(c-a)
Equation of a sphere
|r-a|=R
Equation of a cylinder
|r-(r.n̂)n̂|=R
Shortest distance point to line
d= |(p-a)x(b-a)|/|b-a|
where p is point and a,b are points on line
Shortest distance point to plane
d= |(p-a).n̂|
Where p is point, a is point on plane and n̂ is normal to plane
Shortest distance line to line
d= |(a2-a1).(t1 x t2)|/|(t1 x t2)|
(t1 x t2) is unit vector perpendicular to both lines
