Thread: Determining the equation of a projection

Hi this is the problem
A line x=y=z is projected along a vector
(3,2,1) on the plane : x + z = 0
Determine the equation of the projection

Ok normally a line would be given like this in a basic question:
(x,y,z) = (2+3t,2,6t) for example
However a line with the expression x=y=z
Seems confusing to me

But let me give thoughts on how to solve it
Then maybe someone can help me out

Attempt:

1. To get the projection I will call it
Hum (l)p we Need to know 2 things:

1. A direction vector vp
2. And a point on (l)p

2. A vector is given as : (3,2,1)

A point can be a point of intersection
I will call it S

I will get it by putting the equation of a line
Into the equation of the plane

The problem here is x=y=z
I don't know if I think correctly regarding
This problem

Could someone solve it for me?

1) Line: x = y = z
Parametric form: x = t, y = t, z = t. L(t) = (t, t, t).
Vector form: L(t) = vt, where v = (1, 1, 1) = i + j + k.

2) I am *not* sure what "projecting a line L through a vector w onto a plane P" means. I'll give you my thoughts, but leave it to you, your book, or someone else to provide the exact meaning. Without knowing the exact meaning of what's being asked, you obviously can't solve the problem. I haven't thought this out, so it's possible that two, or even all three, of the following are the same:

Option #1: For each point on L, draw a new line between it and the line formed by the vector w through the origin, so that the new line, and the line formed by w, are perpendicular. Where that newly drawn line intersects P is the projection of that point onto P. When you do that for every single point on L, you'll have projected the line L through w onto P. (Note that this approach breaks down where those two lines intersect.)

Option #2: Intersect P with the plane formed by the line L and the vector w at the origin. The line of intersection between those two planes will be the projection of the line L through w onto P. (Note that this approach exploits that L goes through the origin.)

Option #3: Decompose w = w1 + w2, where w1 and w2 are perpendicular (i.e. orthogonal), and w1 is parallel to L. The idea here is that the w1 part of w doesn't contribute to the projection of L through w, so that "projecting L through w" should mean "take each point of L, and draw the line through it in the w2 direction, and where that line intersects P is the projection of that point through w onto P".

A quick note without double check:

Find the plane containing line x=y=z and vector V={3,2,1}: find a plane prep to V then another prep to that plane and find its direction angles result: plane: x-2y+z=0 this plane contains V & line x=y=z

Now intersect the two planes x-2y+z=0 & x+z=0 to obtain the line of projection of x=y=z on the plane x+z=0...i suspect it projects into line x+z=0?

In this case, it turns out that all 3 are the same regarding the entire line's projection.

All three say that the projection is the line formed by the intersection of P with the plane defined by a normal v x w and containing the point v. (Where v = (1, 1, 1) = i + j + k, w = (3, 2, 1) = 3i + 2j + k.)