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Lec 02 Review of Linear Algebra

Menu of this lecture

Unit vector: \(\hat{a}=\vec{a}/|\vec{a}|\)
Usually use unit vectors to present directions.

Vectors are represented as column vectors by default.

Functions of dot product and cross product

Dot product:

  1. Find the angle between two vectors.
    e.g cosine of angle bwtween light source and surface.
  2. Find projection of one vector on another.

More specifically:

  1. Measure how close two directions are.
  2. Decompose a vector.
  3. Determine forward / backward.

Cross product:

  1. Construction coordinate systems.

Functions:

  1. Determine left / right. Given a plane and two vectors on this plane, determine the relative position of the two vectors.
  2. Determine in / out. Several vectors are connected head-to-tail to form a closed shape. Given another point, determine whether this point lies inside the closed shape.

e.g.
example
Check: \(\vec{AB}\times\vec{AP}, \vec{BC}\times\vec{BP}, \vec{CA}\times\vec{CP}\). If the signs of all three are the same, then point P is inside the shape.

Dot product and cross product in matrix:

2D reflection about y-axis:

\[ \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}= \begin{pmatrix} -x \\ y \end{pmatrix} \]
matrix in latex
Text Only
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\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}

Rendered output:

\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]
\[ \vec{a}\cdot\vec{b}=A^T\cdot B =\begin{pmatrix} x_a & y_a & z_a \end{pmatrix} \begin{pmatrix} x_b \\ y_b \\ z_b \end{pmatrix} \]
\[ \vec{a}\times\vec{b}=A^*B =\begin{pmatrix} 0 & -z_a & y_a \\ z_a & 0 & -x_a \\ -y_a & x_a & 0 \end{pmatrix} \begin{pmatrix} x_b \\ y_b \\ z_b \end{pmatrix} \]

(\(\begin{pmatrix} 0 & -z_a & y_a \\ z_a & 0 & -x_a \\ -y_a & x_a & 0 \end{pmatrix}\) is the dual matrix of \(\vec{a}\))