Equation of plane containing two vectors I am struggling with the interpretation of this question:
Vectors: $u = \left(1,\ 0,\ \sqrt3 \right)$ and $v = (1,\ \sqrt3,\ 0)$ in standard position.
Find an equation of the plane containing $u$ and $v$.
Am I correct in interpreting this question that the plane is parallel to both vectors? Does this mean that I can form the cross product of $u$ and $v$ to find the normal and use one of the vectors as a point on the plane?
Thank you in advance
 A: If you're using the cross product, this gives
$$
\begin{align}
\begin{vmatrix}
e_{\vec{x}} & e_{\vec{y}} & e_{\vec{z}} \\
1 & 0 & \sqrt{3} \\
1 & \sqrt{3} & 0
\end{vmatrix} =
\left(\begin{vmatrix} 0 & \sqrt{3} \\ \sqrt{3} & 0\end{vmatrix},
-\begin{vmatrix} 1 & \sqrt{3} \\ 1 & 0\end{vmatrix},
\begin{vmatrix} 1 & 0 \\ 1 & \sqrt{3}\end{vmatrix}\right) = \left(-3, \sqrt{3}, \sqrt{3} \right)
\end{align}
$$
Thus, $\vec{p} = t(-3,\sqrt{3},\sqrt{3}), \enspace t\neq 0$ is a orthogonal vector to vector $\vec{u}$ and $\vec{v}$ and the plane they make.
If $(x,y,z) and (x_0,y_0,z_0)$ are points on the plane, then $\vec{w} = (x-x_0, y-y_0,z-z_0)$ is a vector on the plane, and $\vec{w}\cdot\vec{p} = 0$:
$$
\begin{align}
t(-3,\sqrt{3},\sqrt{3})\cdot(x-x_0, y-y_0,z-z_0) &= 0 \\
t(-3(x-x_0)+\sqrt{3}(y-y_0)+\sqrt{3}(z-z_0) &= 0
\end{align}
$$
We can choose $t=1$ (because a vector is orthogonal regardless of its length) and $(x_0,y_0,z_0) = (0,0,0)$ (because the plane passes through the origin, because both vectors go through the origin). This yields:
$$-3x+\sqrt{3}y+\sqrt{3}z = 0$$
A: Well, since $u$ and $v$ are not parallel, the plane $P$ is 
$$ P = \{a \cdot u + b \cdot v \} $$
for $a, b \in \mathbb{R}$. So
$$ P = \{(a, 0, \sqrt{3}a) ~ + ~ (b, \sqrt{3}b, 0)  \} = \{(a + b, \sqrt{3}b, \sqrt{3}a)\} $$
To obtain the equation of the plane in Euclidean / Cartesian / xyz-form let 
$$ (a + b, \sqrt{3}b, \sqrt{3}a) = (x, y, z)$$
and observe that $z - \sqrt{3}x + y = 0$.
A: What if you subtract v from u and then use that as the normal (a,b,c) in the equation ax + by + cz = 0
That will give you the equation.
