Cross products in the light of linear transformations

What to learn

  • How can we derive product dot?

Description

  • Before we saw this.

    \[\begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix} \times \begin{bmatrix} w_1 \\ w_2 \\ w_3 \end{bmatrix} = det\left ( \begin{bmatrix} \vec{i} & v_1 & w_1 \\ \vec{j} & v_2 & w_2 \\\vec{k} & v_3 & w_3 \end{bmatrix}\right )\] \[\vec{i}(v_2w_3-v_3w_2) + \vec{j}(v_3w_1-v_1w_3) + \vec{k}(v_1w_2-v_2w_1)\]

  • Let’s think about function \(f(\vec{x})\), derminant of \(\vec{x},\vec{v},\vec{w}\).

    \[f\left ( \begin{bmatrix} x \\ y \\ z \end{bmatrix}\right ) = det\left ( \begin{bmatrix} x & v_1 & w_1 \\ y & v_2 & w_2 \\z & v_3 & w_3 \end{bmatrix}\right ) \cdots (1)\]

    We can get volumn of parallelepiped using function \(f(\vec{x})\). Area of parallelepiped is calculated by (area of parallelogram) X (component of \(\begin{bmatrix} x & y &z \end{bmatrix}^T\) perpendicular to \(\vec{v}\) and \(\vec{w}\)). → \(f(\vec{x})\) is calculated by \(\vec{p} = \begin{bmatrix} v_1 \\ v_2\\ v_3 \end{bmatrix} \times \begin{bmatrix} w_1 \\ w_2 \\ w_3 \end{bmatrix}\) and, do inner dot with \(\begin{bmatrix} x \\ y \\ z \end{bmatrix}\). Then we can express this like this.

    \[f\left ( \begin{bmatrix} x \\ y \\ z \end{bmatrix}\right )=\begin{bmatrix} p_1 & p_2&p_3 \end{bmatrix} \cdot \begin{bmatrix} x \\ y\\ z \end{bmatrix} = p_1x+p_2y+p_3z \cdots(2)\]

    Looking at (1), we can get \(x(v_2w_3-v_3w_2) + y(v_3w_1-v_1w_3) + z(v_1w_2-v_2w_1) \cdots (3)\).

    Looking at (2), we find \(\vec{p}\), \(\begin{matrix} p_1 = v_2 \cdot w_3 - v_3 \cdot w_2 \\ p_2 = v_3\cdot w_1 - v_1 \cdot w_3\\ p_3 = v_1 \cdot w_2 - v_2 \cdot w_1 \end{matrix}\)

    So, we find \(\vec{p}\).

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