# Visualise all vectors perpendicular to one vector and two vectors in R^3 [Strang P19 1.2.6]

(b) The vectors perpendicular to any vector in $\mathbb{R^3}$ lie on what?.
(c) The vectors perpendicular to any two vectors in $\mathbb{R^3}$ lie on what?.

(b) The algebra proves: $\forall \, (a, b, c) \in \mathbb{R^3}$, $(a,b,c) \cdot (x, y, z) = 0 \implies ax + by + cz = 0$, a plane through the origin with normal vector $(a, b, c)$. How do I see this from (descrying) this?

I sketched the blue, green, orange, and red vectors to be parallel to the position vector $p$. Yet I can't see how these four vectors lie on the same plane?

(c) I can see that this is true pictorially, thanks to the cross product. I only need to extend the cross product vector infinitely at both ends to form the line $\perp$ to any two vectors in $\mathbb{R^3}$.

(b) It's $\Bbb R^3$, our dimension. Consider yourself as a vector, then you are perpendicular on what? Of course the flat land that you stand on it, a plane. (Don't forget if the vector is zero then the answer is the whole $\Bbb R^3$.)