Given that if a line L is parallel to a plane P, how can I prove that any direction vector for L must be orthogonal to any normal vector for P?

Question

I know that I need to prove that the direction vector of L dotted with the normal vector of P is 0. However, I am not sure how to show that.

edit: Am I allowed to assume that the direction vector L = k times the linear combination of two vectors since they are parallel?

Answer 1

When the equation of the plane is in $$ax+by+cz=d$$

the normal vector is $N=<a,b,c>$ and all normal vectors to the plane are parallel.

For the equation of a line in parametric form$$ x=x_0 + kt\\ y=y_0 + lt\\z=z_0 + mt$$ the direction vector is $D=<k, l,m>$ and all direction vectors are parallel.

You need to show the dot product of $\alpha D$ and $\beta N$ is zero.