1. ## Quick Vectors Question

I have absolutely no idea what I'm doing with vectors so if someone could help me out with these questions I'd appreciate it. I'm suppose to describe the shape given by these vector equations, where r is a 3 dimensional vector:
r X (i + j) = ( i - j)
|r - i| = |r - j|
For both of these I have solved for x, y and z. For the first one I get z = -1 and y = x. I have solved the second one similarly, however this isn't really describing the shape given by the equations so I'm not sure what is required for an answer.

2. Originally Posted by kevinlightman
I have absolutely no idea what I'm doing with vectors so if someone could help me out with these questions I'd appreciate it. I'm suppose to describe the shape given by these vector equations, where r is a 3 dimensional vector:
r X (i + j) = ( i - j)
|r - i| = |r - j|
For both of these I have solved for x, y and z. For the first one I get z = -1 and y = x. I have solved the second one similarly, however this isn't really describing the shape given by the equations so I'm not sure what is required for an answer.
Let r= xi+yj+ zk. Then "|r- i|= |r- j|" is exactly $\sqrt{(x-1)^2+ y^2+ z^2}= \sqrt{x^2+(y-1)^2+ z^2}$. Squaring both sides, $(x-1)^2+ y^2+ z^2= x^2+ (y-1)^2+ z^2$ or $x^2- 2x+ 1+ y^2+ z^2= x^2- 2y+ 1+ z^2$. Now all squares and the "1"s cancel leaving -2x= -2y or x= y. That is the equation of a plane through the origin: x-y= 0.

Geometrically, if you think of i as representing the point(1, 0, 0) and j as representing the point (0, 1, 0), the set of all points equidistant from them is the plane through the midpoint of the line segment between the two points and perpendicular to it. The midpoint is, of course, the point (1/2, 1/2, 0) and a vector from (1, 0, 0) to (0, 1, 0) is i- j so the plane perpendicular to that and containing (1/2, 1/2, 0) is 1(x- 1/2)- 1(y- 1/2)= x- y= 0 as before.

3. Thanks, I came to that algebraic solution but didn't know how to phrase it in words, so for the first question where x=y and z=-1 would that be a line through (0,0,-1) that is parallel to the xy plane?