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I know that if I have a vector in $R^3$ I can move it around wherever I want, does the same apply to planes? Or are planes constrained to a location? The reason for this questions is that I was given the plane x+3y+4z=8 and asked to find the point closest to the origin on that plane, I have it's parametric equations as well and I thought of setting a variable to zero but Im not sure how its going to work since I keep getting different answers.

x = t +3s y = t - 5s z = 2-t+3s

Teyash Arjun
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    I believe the OP means that the "base point" of a vector in a diagram can be moved yet it is still considered to be the same vector. He seems to ask if we can move a plane defined by a linear equation (with a translation) and consider that to be the same plane. – Rory Daulton Jan 14 '16 at 23:50

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They don't. Planes are sets of points, like lines, cubes, spheres, etc. If you moved them around, you would change the set of points, and so change the plane!

To see this in the algebra, consider a plane:

$$x + y + z = 0$$

This plane goes through the origin (you can tell because $x = 0, y = 0, z = 0$ satisfies the equation). Suppose we wanted to move it up one unit.

We would have to substitute $z \mapsto z - 1$, (for the same reason the line $y = 2(x-1)$ is one unit to the right of $y = 2x$. But when we do this:

$$x + y + z - 1 = 0 \iff x + y + z = 1$$

Which is a different plane.

A vector is not a set of points. Vectors are like directions -- "go up one unit, forward -2 units, and east $\sqrt{2}$ units". Though they can be in the same drawing as a plane, they are fundamentally different objects.

Eli Rose
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