1) Line: x = y = z

Parametric form: x = t, y = t, z = t. L(t) = (t, t, t).

Vector form:L(t) =vt, wherev= (1, 1, 1) =i+j+k.

2) I am *not* sure what "projecting a line L through a vectorwonto a planeP" 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 vectorwthrough the origin, so that the new line, and the line formed byw, are perpendicular. Where that newly drawn line intersectsPis the projection of that point ontoP. When you do that for every single point on L, you'll have projected the line L throughwontoP. (Note that this approach breaks down where those two lines intersect.)

Option #2: IntersectPwith the plane formed by the line L and the vectorwat the origin. The line of intersection between those two planes will be the projection of the line L throughwontoP. (Note that this approach exploits that L goes through the origin.)

Option #3: Decomposew=w1+w2, wherew1andw2are perpendicular (i.e. orthogonal), andw1is parallel to L. The idea here is that thew1part ofwdoesn't contribute to the projection of L throughw, so that "projecting L throughw" should mean "take each point of L, and draw the line through it in thew2direction, and where that line intersectsPis the projection of that point throughwontoP".