Find the scalar equation of the plane through the points M(1,2,3) and N(3,2,-1) that is perpendicular to the plane with equation 3x + 2y + 6z +1 = 0. I know that normal of the latter equation is (3,2,6) but now what do I do? Thanks.
Calculus - Reiny, Monday, July 7, 2008 at 5:23pm
You are on the right track, so (3,2,6) must be a direction vector on your new plane.
you also have two points M(1,2,3) and N(3,2,-1), so the vector MN or (2,0,-1) is another direction vector.
So by taking the cross-product of vectors (3,2,6) and (2,0,-1) I got (2,-6,1)
which must be the normal to the new plane.
So the new plane has equation
2x - 6y + z = k
but M(1,2,3) lies on it, so 2 - 12 - 1 = k
k = -7
The new plane has equation 2x - 6y + x = -7
1. Both M and N satisfy the new equation
2. Is the dot-product of their normals zero? (3,2,6)∙(2,-6,1) = 6-12+6 = 0
There you go!
typo correction - Reiny, Monday, July 7, 2008 at 5:28pm
<but M(1,2,3) lies on it, so 2 - 12 - 1 = k >
should say :
but M(1,2,3) lies on it, so 2 - 12 + 3 = k
Calculus - Lulu, Monday, July 7, 2008 at 9:16pm
I appreciate the help, but you made another mistake. You got a wrong answer, it should be (2,0,-4) not (2,0,-1) so thus, you used the wrong information. But thanks, I know what to do now :)