Representing a Plane as a Vector Equation A general way of representing a plane in $3D$ is, $ax + by+ cd = d$. But in my Calculus 3 course my instructor represented a plane with a vector equation $ a\vec{i} + b\vec{j} +c\vec{k} = d$
Is that a valid representation of a Plane? I mean, of course a plane can represented with vectors like in,  $\vec{r}.\vec{d} = p$ where $\vec{d}$ is a unit normal vector perpendicular to the plane and $p$ is the perpendicular distance of the plane from the origin. In this case, both the left hand side as well as the right hand side is a scalar whereas in the representation used by my instructor the left hand side is a vector while the right hand side is a scalar. It appears a bit disturbing to me.
I can't reach my instructor now. In all the quiz questions, she represented planes as $ a\vec{i} + b\vec{j} +c\vec{k} = d$. Can anyone confirm is it a correct representation? If yes, How one side of the equation is a vector and the other is a scalar?
Thanks a lot in advance
I think the equation your instructor used is incorrect, for the sum of vectors is necessarily a vector.
However we can say the following:
Let $\mathbf{u}$ and $\mathbf{v}$ be any two non-zero, non-collinear vectors in $\mathbb{R}^3$. Then the equation of the plane determined by $\mathbf{u}$ and $\mathbf{v}$ is
$$
\mathbf{x} = \lambda \mathbf{u} + \nu \mathbf{v}, 
$$
where $\lambda, \mu \in \mathbb{R}$ are the variables.
I think the equation your instructor used is incorrect, for the sum of vectors is necessarily a vector.
However we can say the following:
Let $\mathbf{u}$ and $\mathbf{v}$ be any two non-zero, non-collinear vectors in $\mathbb{R}^3$. Then the equation of the plane determined by $\mathbf{u}$ and $\mathbf{v}$ is
$$
\mathbf{x} = \lambda \mathbf{u} + \nu \mathbf{v}, 
$$
where $\lambda, \mu \in \mathbb{R}$ are the variables.