Calculus lll True or false

Question

The plane 3x + 2y − z = 0 is perpendicular to the line x = 3t, y = 2t, z = −t

True or false

Answer 1

The line L parametrized by x = 3t, y = 2t, z = -t can be rewritten as the function L(t) = t <3,2,-1>, showing that the vector <3,2,-1> is parallel to the line.

The plane P expressed via 3x + 2y - z = 0 can be rewritten as <3,2,-1> * <x,y,z> = 0. Two vectors are perpendicular exactly when their dot product is zero. Therefore, in order for the vector <x,y,z> to lie on the plane P, it must follow that is perpendicular to <3,2,-1>. Hence, every vector that lies on plane P will be perpendicular to every vector that lies on line L. Thus, the line L and the plane P are perpendicular.

True.

Answer 2

For any plane, you can find the "normal" (perpendicular) vector to the plane by taking the coefficients of x y and z as a vector, in this case the equation is 3x+2y-z=0 which gives the vector <3,2,-1>. Furthermore, the parametric equations for a line are of the form x=x0+at, y=y0+bt, z=z0+ct. The vector <x0,y0,z0> represents the coordinates of a point on the line, and the vector <a,b,c> represents the direction of the line. In this case x0=y0=z0=0 and the direction vector is <3,2,-1>. So the direction vector of the line is the same vector as the normal vector of the plane which means that the line is perpendicular to the plane.