Question

how do i explain this to a 10th grader who was just introduced to lines and planes

explain why a line in 3-space cannot be represented by a scalar equation like \(ax+by=C\)

Answer 1

Because the "z" component is equal to zero:
\[ax+by+(0)z=C\]
This means that it has "z" coordinate of zero, meaning that it only defines a plane, and that plane will be created by the family of lines:
\[r_1 + k(r_2)=0\]

Answer 2

In some sense you can, the problem is you have to use a system of linear equations to represent a line.

So let's just say the z component depends on x and y, this is true for a line. \(z(x,y)=ax+by+c\) but the problem is this lets us pick any point in the xy plane to a height, which will give us an entire surface! We need to restrict what values we can pick for x and y, so we need to go further and define \(y(x)=mx+n\) which will be the projection of our actual line onto the xy plane, since we are going to map only points from a line to a line on the plane represented by \(z(x,y)\) to get our line.

Answer 3

give her a square and divide it to slices with same ax+by=c equation, so she would figure out its not unique representation.

Answer 4

if you got what i mean