Question

Find an equation of the plane through the point (-2, 2, -1) and parallel to the plane 4x+1y+3z=−1.

Answer 1

if two planes are parallel, what can we say about their normal vectors?

Answer 2

that the vectors are perpendicular?

Answer 3

i don't think so, look at the image
here is the normal vector for one plane \(\vec n_1\), what will the other \(\vec n_2\) look like?

Answer 4

parallel to that vector?

Answer 5

ie n1 is parallel to n2?

Answer 6

correct

Answer 7

now we know that the point (-2,-2,-1) is in the plane we want, so let us draw that

Answer 8

we can represent every other point in our plane as (x,y,z)
now we draw a vector between the known point (-2,2,-1) and any other point in the plane (x,y,z) we will get a vector in the plane

Answer 9

let us call the vector from (x,y,z) to (-2,2,-1) \(\vec v\).
What can we say about \(\vec v\) and \(\vec n\) ?