Find the vectors that span the space orthogonal to a given vector

Question

I want to find the two vectors $v$ and $w$ that span a space orthogonal onto a given vector $A$$(A_x, A_y, A_z)$.

I suppose that the dot-product of v and w should be zero. As well as the dot-products of $v$ and $A$ and the dot-product of $w$ and $A$.

Afterwards I want to project a given vector $b$ onto the plane.

Answer 1

The subspace orthogonal to $A(A_x, A_y, A_z)$ is the plane:

$$xA_x+y A_y+zA_z=0$$

thus is sufficient tha you pick two linearly independent vectors in the plane.

For the projection process take a look here Ways to find the orthogonal projection matrix.