What are the equations for lines and planes in 3D?

[GameProgrammer]

Hi everyone I hope I have the correct category for these questions! (I'm new to the forums). Anyways I'm currently in college studying to become a video game programmer, I've never taken physics before and I was doing fine in my course until we have started learning about Lines and Planes in 3D. I have a few questions regarding this topic. Any help is greatly appreciated! Thanks in advance

Homework Statement

1) Find the parameter equations for the line through the point (5,1,0) that is perpendicular to the plane 2x-y+z=1
2) Find an equation of the plane passing through the point (3,0,8) and parallel to the plane 2x+5y+8z = 17
3)Show that the line through the points (0,1,1) and (1,-1,6) is perpendicular to the line through the points (-4,2,1) and (-1,6,2)

Homework Equations

r = r + tv

x = x + at
y = y + bt
z = z + ct

t = x-xo/a = y-yo/b = z-zo/c

P(t) = t(P1) + (1-t)P2

n*(r-ro) = 0

a(x-xo) + b(y-yo) + c(z-zo)

 

[learningphysics]

Can you post any work that you did, or describe how you tried to approach the problem?

 

[GameProgrammer]

The first two problems I have no idea how to start, but here's what I have on the 3rd
r = r + tv
v = (1-0,-1-1,6-1)
v = <1,2,5>
r1 = <0,1,1>+t<1,2,5>
r1 = <t, 2t+1, 5t+1>

v = (-1-(-4), 6-2, 2-1)
v = <3, 4, 1>
r2 = <-4, 2, 1> + t<3, 4,1>
r2 = <3t-4, 4t+2, t+1>

r1*r2 = 0

 

[learningphysics]

You made a mistake calculating the direction vector of the first line. It should be <1,-2,5>

To check that two lines are perpendicular, you just need to check that their direction vectors are perpendicular... I'm not sure if "perpendicular" means that they must intersect when 3 dimensions are concerned. These two lines don't intersect.

So just check the dot product of the direction vectors.

For the first two problems, think about the normal to the plane. Can you get the normal to the plane, by looking at the equation of the plane? The normal is a line perpendicular to the plane... what can you say about two planes that have the same normal or parallel normals?

 

[uiulic]

The first two problems I have no idea how to start, but here's what I have on the 3rd
r = r + tv
v = (1-0,-1-1,6-1)
v = <1,2,5>
r1 = <0,1,1>+t<1,2,5>
r1 = <t, 2t+1, 5t+1>

v = (-1-(-4), 6-2, 2-1)
v = <3, 4, 1>
r2 = <-4, 2, 1> + t<3, 4,1>
r2 = <3t-4, 4t+2, t+1>

r1*r2 = 0

Your method is not convenient (there are always many different ways to sort out such similar problems). You only need to show (think of slope )

(1-0,-1-1,6-1) and (-1+4,6-2,1) i.e. (1,-2,5) and (3,4,1)

1*3+(-2)*4+1*5=0. This completes the question3.

For others, see learningphysics.

 

[GameProgrammer]

Thanks for the help everyone