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# Determine a Cartesian equation for given planes

For each of the following planes, find a linear Cartesian equation of the form

that describes the plane.

1. The plane through spanned by .
2. The plane through the points .
3. The plane through the point parallel to the plan through spanned by and .

1. The plane through spanned by and is the set of points

Therefore, we have the parametric equations

Then we want to solve for in terms of . From the first equation we have

From the second equation we then have

Which gives us

So, from the third equation we then have

Thus,

is the requested linear Cartesian equation.

2. The plane through the three points is the set of points

But, and are in the linear span of since

Thus, this plane is equal to the plane in part (a). Hence, we have the linear Cartesian equation,

3. Again, this is the same plane as in parts (a) and (b) since the span of and is the same as the span of and . Hence, the requested linear Cartesian equation is