Cross product

  1. What to learn
    • What is the Cross product
  2. Description
    • Notation
      • \[\vec{x} \times \vec{y} = \vec{z}\]
    • New vector that length is same the parallelagram that two vectors make. It’s direction is decided by right hand rule.
    • How to calculate
      • Determinant is needed.
        • About two vector \(\vec{x} = \begin{bmatrix} x_1 & x_2 \end{bmatrix}\), \(\vec{y} = \begin{bmatrix} y_1 & y_2 \end{bmatrix}\), we can think it as transformation \(\begin{bmatrix} x_1 & y_1 \\ x_2 & y_2 \end{bmatrix}\). And find how square area that basis vector make is changed.
    • \(\vec{z}\) is bigger when two vector \(\vec{x}\) and \(\vec{y}\) is perpendicular.
    • \(\vec{z}\) is linear about length of two vector \(\vec{x}\) and \(\vec{y}\).
    • To make it more general, \(\vec{x} = \begin{bmatrix} x_1 & x_2 & x_3 \end{bmatrix}\) and \(\vec{y} = \begin{bmatrix} y_1 & y_2 & y_3 \end{bmatrix}\). So,

    • \(\vec{x}\times\vec{y} = \vec{z}\) is

      \[\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \times \begin{bmatrix} y_1 \\ y_2 \\ y_3 \end{bmatrix} = \begin{bmatrix} x_2y_3 - x_3y_2 \\ x_3y_1-x_1y_3 \\ x_1y_2-x_2y_1 \end{bmatrix}\]

      next step, we will talk about why this calculation appears.

  3. Next Step
    • Cross products in the light of linear transformations
  4. References