Vectors

Question 1

Spherical polar coordinates

Answer 1

x=rsin(theta)cos(phi)
y=rsin(theta)sin(phi)
z=rcos(theta)

Question 2

Cylindrical polar coordinates

Answer 2

x=rcos(theta)
y=rsin(theta)
z=z

Question 3

a x (b x c)

Answer 3

(a . c)b - (a . b)c

Question 4

(a x b) x c

Answer 4

(a . c)b - (b.c)a

Question 5

Scalar triple product

Answer 5

a . b x c

Question 6

Scalar product

Answer 6

a . b = mag(a)mag(b)cos(theta)

Question 7

Vector product

Answer 7

a x b = mag(a)mag(b)sin(theta)

Question 8

Equation of a line

Answer 8

r = a + k(b-a)

[get on to line and then move along]

Question 9

Component of a in n̂

Answer 9

a.n̂

Question 10

Projection of a in n̂

Answer 10

(a.n̂)n̂

[a.n̂ gives number - multiply by vector to give projection]

Question 11

Equation of plane (point and normal)

Answer 11

r.n̂=a.n̂

Question 12

Perpendicular distance to origin

Answer 12

r.n̂=a.n̂=p

Question 13

Equation of a plane (3 points)

Answer 13

r-a=mu(b-a)+ upsilon(c-a)

Question 14

Equation of a sphere

Answer 14

|r-a|=R

Question 15

Equation of a cylinder

Answer 15

|r-(r.n̂)n̂|=R

Question 16

Shortest distance point to line

Answer 16

d= |(p-a)x(b-a)|/|b-a|

where p is point and a,b are points on line

Question 17

Shortest distance point to plane

Answer 17

d= |(p-a).n̂|

Where p is point, a is point on plane and n̂ is normal to plane

Question 18

Shortest distance line to line

Answer 18

d= |(a2-a1).(t1 x t2)|/|(t1 x t2)|

(t1 x t2) is unit vector perpendicular to both lines