Generalizing the geometric interpretation of dot product to simple $k$-vectors

Question

Background: For $u, v \in \mathbb R^n$, the dot product $u \cdot v$ can be interpreted geometrically as follows:

1. Its magnitude is the product of the lengths of $u$ and $\operatorname{proj}_{u} v$.
2. Its sign is $+1$ if $u$ and $\operatorname{proj}_{u} v$ point in the same direction, and $-1$ if they point in opposite directions.

Similarly, here is a geometric interpretation for the dot product of two simple $k$-vectors. Recall that the (standard) dot product in $\Lambda^k(\mathbb R^n)$ is given by
$$(u_1 \wedge \cdots \wedge u_k) \cdot (v_1 \wedge \cdots \wedge v_k) = \det(u_i \cdot v_j)_{i,j=1}^k.$$

Let $P(u_1, \ldots, u_k)$ be the (oriented) $k$-dimensional parallelogram generated by $u_1, \ldots, u_k$. Then it seems to me that $(u_1 \wedge \cdots \wedge u_k) \cdot (v_1 \wedge \cdots \wedge v_k)$ can be interpreted geometrically as follows:

1. Its magnitude is the product of the $k$-dimensional volumes of of $P(u_1, \ldots, u_k)$ and $\operatorname{proj}_{\operatorname{span}(u_1, \ldots, u_k)} P(v_1, \ldots, v_k)$.
2. Its sign is $+1$ if $P(u_1, \ldots, u_k)$ and $\operatorname{proj}_{\operatorname{span}(u_1, \ldots, u_k)} P(v_1, \ldots, v_k)$ have the same orientation, and $-1$ if they have opposite orientations.

(To prove this, first note that it is true when $u_1, \ldots, u_k$ are orthonormal. Then use Gram-Schmidt to reduce to the orthonormal case.)

Question: After searching on the internet and in various textbooks, I cannot find this geometric description anywhere. (I did see it briefly mentioned in the question in Interpreting the determinant of matrices of dot products, but it was not addressed in the answer.) That is surprising to me because it seems like a natural generalization of the $k=1$ geometric interpretation, which (I believe) is commonly taught. Is there a textbook or reference that contains the interpretation above for $k > 1$? (Or, did I make a mistake somewhere?)

Answer 1

I found this geometric description in Chapter 3 of David Bachman's A Geometric Approach to Differential Forms (2nd edition).