9.4 Intersection of Three Planea

Question 1

without solving, how can you tell a system of three planes do not intersect?

Answer 1

if you substract two equations and they give 0x+0y+0x+d=0

if two equations in the system have the same coefficients

Question 2

how to calculate the distance between two lines in vector equation form?

Answer 2

1. write parametric equations
   2.choose a point on one of the equations, point A
2. choose a point on other line perpendicular to point, point C
3. for direction vector for line AC
   5.calculate dot product of direction vector and AC and solve for t
4. determine point C by plugging t value in
   7.calculate distance between point A and C

Question 3

how to calculate the distance between two parallel planes?

Answer 3

1. state the direction vectors for each of the planes
2. find a point for one of the planes
3. substitute use the other plane and the point in the distance between two planes equation