I need to ask a question about vectors and cross product? When you take the determinant on 3 vectors, you calculate and get the volume of that specific shape, correct?
When you take the cross-product of 2 vectors, you calculate and get the area of that shape and you also get the vector perpendicular to the plane, correct?
 A: Here is the connection: The cross product can be defined as the unique vector $a \times b$ that satisfies $\langle x, a \times b \rangle = \det \begin{bmatrix} a & b & x \end{bmatrix}$ for all $x$.
The (signed) volume spanned by $a,b,x$ is given by $\det \begin{bmatrix} a & b & x \end{bmatrix}$.
It is easy to see from the formula (let $x = a,b$) that $a \times b$ is perpendicular to both $a$ and $b$, and that the volume spanned by $a,b,a \times b$ is given by $V = \langle a \times b, a \times b \rangle = \|a \times b \|^2$. If $A$ is the area spanned by $a$ and $b$, then we see that $V = \|a \times b \| A$, from which we see that $A = \frac{V}{\|a \times b \|} = \|a \times b \|$.
A: Kind Of.
When you take the determinant of a set of vectors, you get the volume bounded by the vectors.
For instance, the determinant of the identity matrix (which can be considered as a set of vectors) gives the volume of the solid box in $n$ dimensions. A $3\times3$ identity matrix gives the area of a cube.
However, when you calculate cross products, the matrix of whose determinant you take has the first row to be the the unit vectors in the $n$ dimensions.
For instance
\begin{align}
\det\begin{pmatrix}
\hat{i}&\hat{j}&\hat{k}\\1&0&0\\0&1&0
\end{pmatrix}=1 \hat{k}
\end{align}
It does NOT return a scalar value.
A: The absolute value of the determinant having rows the components on Ox, Oy, Oz axis is indeed the volume of the parallelipeped spanned by the 3 vectors.
When you take the cross product you get a vector perpendicular on the plane containing the inial 2 vectors and with direction given by the right-hand rule.
If you take the magnitude of this cross-product you get the area of the parallelogram spanned by the 2 vectors.
So cross-product gives a vector. Area is scalar (magnitude of vector)
