How to Find the Normal Vector for a Plane Perpendicular to Another?

[eme_girl]

Find the scalar eq'n of a plane that is perpendicular to the plane with normal vector [3,1,2] and passes through points A(2,-6,-1) and B(1,2,-4).

I think that the normal vector can be the direction vector of this new plane. But then, in order to find the scalar eq'n I need a normal vector of this new plane. How do I find this?

 

[HallsofIvy]

Strictly speaking a plane doesn't have a "direction vector". What is true is that the vector [3,1,2] is in the plane you want. You also know that the vector from A(2,6,-1) to B(1,2,-4) (which is, of course, [1-2,2-6,-4-(-1)]= [-1, -4, -3] is in the plane. Do you know how to find a vector that is perpendicular to both [3,1,2] and [1,4,-3]?

 

[eme_girl]

I understand what you just found. But no, I do not know how to find a vector's that perpendicular to both those vectors.

 

[TenaliRaman]

eme_girl,
do u know the direction of a vector that is a cross product of two vectors?

-- AI

 

[HallsofIvy]

TenaliRaman's point: the cross product of two vectors is always perpendicular to both.

The cross product of [a₁,a₂,a₃] and [b₁,b₂,b₃] is the vector [a₂b₃-a₃b₂,a₃b₁-a₁b₃,a₁b₂-a₂b₁].