Geometry

p1

p2

p3

Unit Vector (grade 1)

$$e_1\quad,e_2\quad,e_3\quad,e_4\quad\dots$$

depending on the dimension used

Unit Bivector (grade 2)

2D : $$\quad e_1e_2$$

3D : $$\quad e_1e_2,\quad e_1e_3,\quad e_2e_3$$

4D : $$\quad e_1e_2,\quad e_1e_3,\quad e_1e_4,\quad e_2e_3,\quad e_2e_4,\quad e_3e_4$$

etc. depending on dimension used

Unit Trivector (grade 3)

3D : $$\quad e_1e_2e_3$$

4D : $$\quad e_1e_2e_3,\quad e_1e_2e_4,\quad e_1e_3e_4,\quad e_2e_3e_4$$

etc. depending on dimension used




2D Setup : $$\overset{\text{scalar}}{{1}}\quad\quad\overset{\text{vector}}{\overbrace{e_1\quad\quad e_2}}\quad\quad \overset{\text{bivector}}{\overbrace{e_1e_2}}$$

$$a=a_1e_1 + a_2e_2$$

$$b=b_1e_1 + b_2e_2$$

Addition

$$\begin{array}{ll}a+b&=(a_1+b_1)e_1+(a_2+b_2)e_2\end{array}$$

Multiplication (Product) Warning : not commutative

$$\begin{array}{ll}ab&=(a_1b_1 + a_2b_2) + (a_1b_2 - a_2b_1)e_1e_2\end{array}$$

$$\begin{array}{ll}ab
&=(a_1e_1 + a_2e_2)(b_1e_1 + b_2e_2) \\ \\
&=a_1e_1(b_1e_1 + b_2e_2) + a_2e_2(b_1e_1 + b_2e_2) \\ \\
&=a_1b_1e_1e_1 + a_1b_2e_1e_2 + a_2b_1e_2e_1 + a_2b_2e_2e_2 \\ \\
&=a_1b_1 + a_1b_2e_1e_2 - a_2b_1e_1e_2 + a_2b_2 \\ \\
&=(a_1b_1 + a_2b_2) + (a_1b_2 - a_2b_1)e_1e_2 \\ \\ \end{array}$$

Magnitude (Absolute, Modulus, Norm)

$$\begin{array}{ll}\|a\|=\sqrt{\overset{\phantom{0}}{aa}}&=\sqrt{\overset{\phantom{0}}{{{a_1}}^2 + {{a_2}}^2}}\end{array}$$

$$\begin{array}{ll}aa
&=(a_1a_1 + a_2a_2) + (a_1a_2 - a_ab_1)e_1e_2 \\ \\
&={{a_1}}^2 + {{a_2}}^2 \\ \\
\sqrt{\overset{\phantom{0}}{aa}}&=\sqrt{\overset{\phantom{0}}{{{a_1}}^2 + {{a_2}}^2}} \\ \\ \end{array}$$

Direction (Signum, Normalization, Unit)

$$\begin{array}{lll}\hat{a}&=\dfrac{a}{\sqrt{\overset{\phantom{0}}{aa}}}&=\dfrac{a_1\sqrt{\overset{\phantom{0}}{{{a_1}}^2 + {{a_2}}^2}}}{{{a_1}}^2 + {{a_2}}^2}e_1 +\dfrac{a_2\sqrt{\overset{\phantom{0}}{{{a_1}}^2 + {{a_2}}^2}}}{{{a_1}}^2 + {{a_2}}^2}e_2 \end{array}$$

$$\begin{array}{lll}\dfrac{a}{\sqrt{\overset{\phantom{0}}{aa}}} &=\dfrac{a}{\sqrt{\overset{\phantom{0}}{{{a_1}}^2 + {{a_2}}^2}}}\\ \\&=\dfrac{a_1\sqrt{\overset{\phantom{0}}{{{a_1}}^2 + {{a_2}}^2}}}{{{a_1}}^2 + {{a_2}}^2}e_1 + \dfrac{a_2\sqrt{\overset{\phantom{0}}{{{a_1}}^2 + {{a_2}}^2}}}{{{a_1}}^2 + {{a_2}}^2}e_2 \\ \\ \end{array}\\ \\$$

Polar Form

$$\begin{array}{lll}v&=v_1e_1 +v_2e_2&=\|v\|\hat{v}\end{array}$$


Sandwich Product

$$\begin{array}{ll}vav&=\Big(({{v_1}}^2 - {{v_2}}^2) a_1 + 2v_1v_2a_2\Big)e_1 +\Big(({{v_2}}^2 - {{v_1}}^2) a_2 + 2v_1v_2a_1\Big)e_2\end{array}$$

$$\begin{array}{ll}vav&=(v_1e_1 +v_2e_2)(a_1e_1 +a_2e_2)(v_1e_1 +v_2e_2)\\ \\&=((v_1a_1 + v_2a_2) + (v_1a_2 - v_2a_1)e_1e_2)(v_1e_1 +v_2e_2) \\ \\&=(v_1a_1 + v_2a_2)(v_1e_1 +v_2e_2) + (v_1a_2 - v_2a_1)e_1e_2(v_1e_1 +v_2e_2) \\ \\&=(v_1a_1 + v_2a_2)(v_1e_1 +v_2e_2) + (v_1a_2 - v_2a_1)(-v_1e_2 +v_2e_1) \\ \\&=v_1(v_1a_1 + v_2a_2)e_1 +v_2(v_1a_1 + v_2a_2)e_2 -v_1(v_1a_2 - v_2a_1)e_2 +v_2(v_1a_2 - v_2a_1)e_1 \\ \\&=(v_1v_1a_1 + v_1v_2a_2)e_1 +(v_2v_1a_1 + v_2v_2a_2)e_2 +(-v_1v_1a_2 +v_1v_2a_1)e_2 +(v_2v_1a_2 - v_2v_2a_1)e_1 \\ \\&=(v_1v_1a_1 + v_1v_2a_2+v_2v_1a_2 - v_2v_2a_1)e_1 +(v_2v_1a_1 + v_2v_2a_2-v_1v_1a_2 +v_1v_2a_1)e_2 \\ \\&=({{v_1}}^2 a_1 - {{v_2}}^2 a_1 + 2v_1v_2a_2)e_1 +({{v_2}}^2 a_2 - {{v_1}}^2 a_2 + 2v_1v_2a_1)e_2\\ \\&=\Big(({{v_1}}^2 - {{v_2}}^2) a_1 + 2v_1v_2a_2\Big)e_1 +\Big(({{v_2}}^2 - {{v_1}}^2) a_2 + 2v_1v_2a_1\Big)e_2\end{array}$$

This multiplication in polar

$$\begin{array}{ll}vav&=\|v\|\hat{v}\|a\|\hat{a}\|v\|\hat{v}\\ \\ &=\|v\|\|a\|\|v\|\hat{v}\hat{a}\hat{v}\end{array}$$

let

$$\begin{array}{ll}\hat{v}&=\hat{v}_1 e_1 + \hat{v}_2 e_2\\ \\\hat{a}&=\hat{a}_1 e_1 + \hat{a}_2 e_2\\ \\ \end{array}$$

$$\begin{array}{ll}\hat{v}\hat{a}\hat{v}&=\Big(({{\hat{v}_1}}^2 - {{\hat{v}_2}}^2) \hat{a}_1 + 2\hat{v}_1\hat{v}_2\hat{a}_2\Big)e_1 +\Big(({{\hat{v}_2}}^2 - {{\hat{v}_1}}^2) \hat{a}_2 + 2\hat{v}_1\hat{v}_2\hat{a}_1\Big)e_2 \end{array}$$

notice that this operation happens on unit circle as all magnitude left out, this operation does not change




3D Setup : $$\overset{\text{scalar}}{{1}}\quad\quad\overset{\text{vector}}{\overbrace{e_1\quad\quad e_2\quad\quad e_3}}\quad\quad \overset{\text{bivector}}{\overbrace{e_1e_2}\quad\quad\overbrace{e_1e_3}\quad\quad\overbrace{e_2e_3}}\quad\quad\overset{\text{trivector}}{{e_1e_2e_3}}$$

$$a=a_1e_1 + a_2e_2 + a_3e_3$$

$$b=b_1e_1 + b_2e_2 + b_3e_3$$

Addition

$$\begin{array}{ll}a+b&=(a_1+b_1)e_1+(a_2+b_2)e_2+(a_3+b_3)e_3\end{array}$$

Multiplication (Product) Warning : not commutative

$$\begin{array}{ll}ab&=(a_1b_1 + a_2b_2 + a_3b_3) + (a_1b_2 - a_2b_1)e_1e_2 + (a_1b_3 - a_3b_1)e_1e_3 + (a_2b_3 - a_3b_2)e_2e_3\end{array}$$

$$\begin{array}{ll}ab&=(a_1e_1 + a_2e_2 + a_3e_3)(b_1e_1 + b_2e_2 + b_3e_3) \\ \\
&=a_1e_1(b_1e_1 + b_2e_2 + b_3e_3) + a_2e_2(b_1e_1 + b_2e_2 + b_3e_3) + a_3e_3(b_1e_1 + b_2e_2 + b_3e_3) \\ \\
&=a_1b_1e_1e_1 + a_1b_2e_1e_2 + a_1b_3e_1e_3 + a_2b_1e_2e_1 + a_2b_2e_2e_2 + a_2b_3e_2e_3 + a_3b_1e_3e_1 + a_3b_2e_3e_2 + a_3b_3e_3e_3 \\ \\
&=a_1b_1 + a_1b_2e_1e_2 + a_1b_3e_1e_3 - a_2b_1e_1e_2 + a_2b_2 + a_2b_3e_2e_3 - a_3b_1e_1e_3 - a_3b_2e_2e_3 + a_3b_3 \\ \\
&=(a_1b_1 + a_2b_2 + a_3b_3) + (a_1b_2 - a_2b_1)e_1e_2 + (a_1b_3 - a_3b_1)e_1e_3 + (a_2b_3 - a_3b_2)e_2e_3 \\ \\ \end{array}$$

Magnitude (Absolute, Modulus, Norm)

$$\begin{array}{ll}\|a\|=\sqrt{\overset{\phantom{0}}{aa}}&=\sqrt{\overset{\phantom{0}}{{{a_1}}^2 + {{a_2}}^2 + {{a_3}}^2}}\end{array}$$

$$\begin{array}{ll}aa &=(a_1a_1 + a_2a_2 + a_3a_3) + (a_1a_2 - a_2a_1)e_1e_2 + (a_1a_3 - a_3a_1)e_1e_3 + (a_2b_3 - a_3a_2)e_2e_3 \end{array}$$

Direction (Signum, Normalization, Unit)

$$\begin{array}{lll}\hat{a}&=\dfrac{a}{\sqrt{\overset{\phantom{0}}{aa}}}&=\dfrac{a_1\sqrt{\overset{\phantom{0}}{{{a_1}}^2 + {{a_2}}^2 + {{a_3}}^2}}}{{{a_1}}^2 + {{a_2}}^2 + {{a_3}}^2}e_1 +\dfrac{a_2\sqrt{\overset{\phantom{0}}{{{a_1}}^2 + {{a_2}}^2 + {{a_3}}^2}}}{{{a_1}}^2 + {{a_2}}^2 + {{a_3}}^2}e_2 +\dfrac{a_3\sqrt{\overset{\phantom{0}}{{{a_1}}^2 + {{a_2}}^2 + {{a_3}}^2}}}{{{a_1}}^2 + {{a_2}}^2 + {{a_3}}^2}e_3 \end{array}$$

Polar Form

$$\begin{array}{lll}v&=v_1e_1 +v_2e_2 +v_3e_3&=\|v\|\hat{v}\end{array}$$

Outer Product, Warning : anti-commutative

$$\begin{array}{ll}\dfrac{1}{2}\Big(ab-ba\Big)&=(a_1b_2 - a_2b_1)e_1e_2 + (a_1b_3 - a_3b_1)e_1e_3 + (a_2b_3 - a_3b_2)e_2e_3\end{array}$$

$$\begin{array}{ll}ab&=(a_1b_1 + a_2b_2 + a_3b_3) + (a_1b_2 - a_2b_1)e_1e_2 + (a_1b_3 - a_3b_1)e_1e_3 + (a_2b_3 - a_3b_2)e_2e_3 \\ \\ ba&=(b_1a_1 + b_2a_2 + b_3a_3) + (b_1a_2 - b_2a_1)e_1e_2 + (b_1a_3 - b_3a_1)e_1e_3 + (b_2a_3 - b_3a_2)e_2e_3 \\ \\ &=(b_1a_1 + b_2a_2 + b_3a_3) - (a_1b_2 - a_2b_1)e_1e_2 - (a_1b_3 - a_3b_1)e_1e_3 - (a_2b_3 - a_3b_2)e_2e_3  \\ \\ ab-ba &=2 (a_1b_2 - a_2b_1)e_1e_2 +2(a_1b_3 - a_3b_1)e_1e_3 +2(a_2b_3 - a_3b_2)e_2e_3 \\ \\ \dfrac{1}{2}\Big(ab-ba\Big) &=(a_1b_2 - a_2b_1)e_1e_2 +(a_1b_3 - a_3b_1)e_1e_3 +(a_2b_3 - a_3b_2)e_2e_3 \\ \\ \end{array}$$

M






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