The basic elements

There are only a few basic elements we can manipulate:

  • The point p~\tilde pp~, which is a concrete physical point, cannot be represented numerically in the absence of a reference frame.
  • The vector v⃗\vec vv, which represents the motion from point to point, cannot be expressed numerically in the absence of a reference frame.
  • The coordinate vector v, the number object of the real numbers, the vector is placed in some coordinate system, and it can be represented numerically.
  • The coordinate system W⃗t\vec W^tW t is represented by an origin and three bidirectionally orthogonal unit vectors.

Operations consisting of the above elements have and only the following operations:

  1. V ⃗\vec vv = p~ 1tilde p_1p~ 1-p ~ 2tilde p_2p~2
  2. The -v ⃗ \ vec vv
  3. A v ⃗ \ vec vv
  4. V ⃗1\vec v_1v 1 + v⃗2\vec v_2v 2
  5. P ~\tilde pp~ + v⃗\vec vv
  6. V ⃗1\vec v_1v 1· v⃗2\vec v_2v 2
  7. V ⃗1\vec v_1v 1× v⃗2\vec v_2v 2

Note: The coordinates of point p~\tilde pp~ are [147] \left[\begin{matrix}1 \\ 4\7\end{matrix} right]⎣⎢, “147 \ ⎥⎤”, “only in a certain coordinate system can be expressed with specific numbers. The coordinates of point p~\tilde pp~ under the coordinate system W⃗t\vec W^tW t are [147] \left[\begin{matrix}1 \\ 4 \\7 \end{matrix} right]⎣⎢, “147” and “⎥⎤”, and the vectors are the same.

Review of matrix operations

  1. If the matrix N is equal to M, then N has the same dimension (row and column) as M and the corresponding elements are equal.
  2. An M by N matrix is M rows (horizontal) and N columns (vertical)

[ a 11 … a 1 n   ⋮ ⋮ a m 1 … a m n ] \left[ \begin{matrix}a_{11} & \cdots & a_{1n} \\\ \vdots & & \vdots \\ a_{m1} & \cdots & a_{mn} \end{matrix}\right]
  1. Matrix multiplication

C = A B = [ 1 2 3   4 5 6 ] [ 1 4 2 5 3 6 ] = [ 1 x 1 + 2 x 2 + 3 x 3 1 x 4 + 2 x 5 + 3 x 6 4 x 1 + 5 x 2 + 6 x 3 4 x 4 + 5 x 5 + 6 x 6 ] = [ 14 32 32 77 ] C = AB = \left[ \begin{matrix}1 & 2 & 3 \\\ 4 & 5 & 6\end{matrix}\right] \left[ \begin{matrix}1 & 4 \\ 2 & 5 \\ 3 & 6 \ end {matrix} \ \ \ = \ left right] [\ begin {matrix} 1 * 1 + 2 * 2 + 3 * 3 & 1 * 4 + 2 x 5 + 3 * 6 \ \ 4 * 1 + 5 * 2 + 6 * 3 & 4 * 4 + 5 * 5 + 6 x6 \end{matrix}\right] = \left[ \begin{matrix}14 & 32 \\ 32 & 77 \end{matrix}\right]