Understand Cross Product

Let two vectors be defined as:

$$\vec{v} = (v_1, v_2, v_3)$$ $$\vec{w} = (w_1, w_2, w_3)$$

The Cross Product as a Linear Map

We define an unknown vector $\vec{n} = [n_x \ n_y \ n_{z}]$ such that its dot product with any vector $\vec{r} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$ equals the determinant of the matrix formed by $\vec{r}, \vec{v}, \text{and } \vec{w}$:

$$\underbrace{[n_x \ n_y \ n_z]}_{\vec{n}^T} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \det(\vec{r}, \vec{v}, \vec{w}) = \begin{vmatrix} x & v_1 & w_1 \\ y & v_2 & w_2 \\ z & v_3 & w_3 \end{vmatrix}$$

Here, the vector $\vec{n}$ representing this transformation is the Cross Product ($\vec{v} \times \vec{w}$).

Geometric Interpretation

To find the components of this “Target Vector” (the normal vector), we typically expand using basis vectors $\mathbf{i}, \mathbf{j}, \mathbf{k}$:

$$\vec{n} = \begin{vmatrix} \mathbf{i} & v_1 & w_1 \\ \mathbf{j} & v_2 & w_2 \\ \mathbf{k} & v_3 & w_3 \end{vmatrix}$$

Understanding “Height” and “Area”

The geometric meaning of the determinant (Scalar Triple Product) is the Volume of the Parallelepiped spanned by $\vec{r}$, $\vec{v}$, and $\vec{w}$.

$$\text{Volume} = \text{Base Area} \times \text{Height}$$

• Base Area: The magnitude of the cross product, $\|\vec{v} \times \vec{w}\|$, represents the area of the parallelogram formed by $\vec{v}$ and $\vec{w}$.
• Height: The projection of vector $\vec{r}$ onto the normal vector $\vec{n}$.

The Plane Equation

It follows that if the determinant is zero, the volume is zero.

$$\begin{vmatrix} x & v_1 & w_1 \\ y & v_2 & w_2 \\ z & v_3 & w_3 \end{vmatrix} = 0$$

Geometric Conclusion: If the volume is 0, the “height” is 0. This means the vector $\vec{r} = (x, y, z)$ lies flat in the same plane spanned by $\vec{v}$ and $\vec{w}$.

Thus, this equation defines the plane passing through the origin spanned by vectors $\vec{v}$ and $\vec{w}$.