Explain why the line (x,y,z) = (4,2,1) + t(2,3,4) is perpendicular to 3x+4y+5z = 10.
I know that this is vector form of an equation, well actually I have no clue how to find perpendicular vectors. Something to do with dot product equaling to 0. help?
First, the problem does NOT ask you to find a perpendicular. It only asks you to verify that the given line is perpendicular to the given plane. You should know two things before you attempt such a problem:
1) If a plane is given in the form Ax+ By+ Cz= D, then the vector <A, B, C> is perpendicular to the plane.
2) If a line is given in the form (a, b, c)+ t(X, y, Z) then the vector <X, Y, Z> is in the same direction as the line.
Those two together say that if line (a, b, c)+ t(X, Y, Z) is perpendicular to the plane Ax+ By+ Cz= D, then the vectors <A, B, C> and <X, Y, Z> are parallel and so one is a multiple of the other. In your case, <A, B, C> and <X, Y, Z> are the same. The "multiple" is 1.