Question

Write the equation of the plane P with vector equation <1,4,-3> . = 7 in the form a(x-x0) + b(y-y0) + c(z-z0) = 0.

Answer 1

"Simplify" that dot product like so: \(\langle1,4,-3\rangle\cdot\langle x,y,z\rangle=1x+4y-3z\)

Answer 2

once you do that you just need to manipulate it so the right-hand side is 0

Answer 3

So I could do x=7, y=0, z=0?

Answer 4

Yes, of course :-) there are an infinite number of possibilities for \((x_0,y_0,z_0)\) -- it only need be a point on our plane

Answer 5

Similarly, \((-1,2,0)\) would also work. You can rewrite our right-hand side in the form \(7=\langle1,4,-3\rangle\cdot\langle7,0,0\rangle\) so that we have:$$\langle1,4,-3\rangle\cdot\langle x,y,z\rangle=\langle1,4,-3\rangle\cdot\langle7,0,0\rangle\\\langle1,4,-3\rangle\cdot\langle x,y,z\rangle-\langle1,4,-3\rangle\cdot\langle7,0,0\rangle=0\\\langle1,4,-3\rangle\cdot\langle x-7,y,z\rangle=0\\(x-7)+4y-3z=0$$