<jupyter_start><jupyter_text>G(2)>  bivectors: grade 2, pseudo vector>>  $ e_1e_2,\quad (e_1e_2)^2 = -1 = I$>  vectors: grade 1>>  $e_1, e_2, (e_1)^2=1, (e_2)^2 = 1,\quad  e_1e_2 = e_1 \wedge e_2,\quad e_1e_1 = e_1 \cdot e_1 = 1$>  scalars: grade 0>>  1>  dot product>>  $ u \dot v = |u||v|cos\theta = \frac{1}{2}(uv + vu)$>  wedge product>>  $ u \wedge v = |u||v|sin\theta = \frac{1}{2}(uv - vu)\\e_1 \wedge e_2 = e_1e_2 = -e_2 \wedge e_1$>  geometric product>>  $ uv = |u||v|e^{I\theta} = u \cdot v + u \wedge v \\(uv)w = u(vw),\quad u(v+w) = uv + uw,\quad u^2 = uu = u \cdot = |u|^2, \quad uv = -vu $  G(3) $\epsilon_{ijk} $>  trivector: grade 3: pseudoscalar>>  $ e_1e_2e_3 = e_1 \wedge e_2 \wedge e_3 = volume = I \therefore (e_1e_2e_2)^2 = -1 \\I^{-1} = -I \\\because (e_1e_2e_3)I^{-1} = 1 \to (e_1e_2e_3)(-e_1e_2e_3) \to-\big((e_1e_2e_3)(e_1e_2e_3)\big) = 1(-1) = 1$>  bivectors: grade 2 : pseudovector: axial vectors>>  $e_1e_2, e_2e_3, e_3e_1 \\e_1e_2 = - e_2e_1$>  vectors: grade 1: polar vectors>>  $ e_1, e_2, e_3 $>  scalar: grade 0>>  1<jupyter_code>ipy.display.YouTubeVideo('https://www.youtube.com/watch?v=njxg5nUwZKA&list=PLpzmRsG7u_gqaTo_vEseQ7U8KFvtiJY4K&index=10',width=800,height=480)<jupyter_output><empty_output><jupyter_text>Bivectors Multiplication Bivector can be made by $\color{red}{( u \wedge v )} = \color{red}{(uv - vu)} $ | $ u,v = vectors $>  $ aB =\;  _1 + _3 \;= a \cdot B + a \wedge B, \quad Ba =\; _1 + _3 \\\therefore \;_1 \;=\; -_1, \quad _3 \;=\; _3 \\\quad \;_0 \;=\; _0, \quad _2 \;=\; -_2 \\\quad \;_0 \;=\; -_0, \quad _1 \;=\; _1 \\\therefore aB - Ba = 2_1=2(a\cdot B), \\    \therefore aB + Ba = 2_3 = 2(a \wedge B) \\\quad uv - vu = 2(u \wedge v) , \quad uv + uv = 2(u \cdot v)\quad q^v_1q^v_2 - q^v_2q^v_1 = 2(q^v_1 \times q^v_2) , \quad q^v_1q^v_2 + q^v_2q^v_1 = -2(q^v_1 \cdot q^v_2) \\\therefore _1 = \frac{1}{2}\big(aB - Ba \big) = a \cdot B \\\quad _0 = \frac{1}{2}\big(uv + vu \big)= u \cdot v \\\quad _0 = \frac{1}{2}\big(q^v_1q^v_2 + q^v_2q^v_1 \big)= -(q^v_1 \cdot q^v_2) \\\therefore _3 = \frac{1}{2}\big(aB + Ba \big) = a \wedge B \\\quad _2 = \frac{1}{2}\big(uv - vu \big)= u \wedge v \\\quad _1 = \frac{1}{2}\big(q^v_1q^v_2 - q^v_2q^v_1 \big)= q^v_1 \times q^v_2$     --->>  quaternion>>>  $ q^v_1 q^v_2 = -(q^v_1 \cdot q^v_2) + q^v_1 \times q^v_2$>>>  $ q^v_1 q^v_2 - q^v_2q^v_1 = 2(q^v_1 \times q^v_2)$>>>  $ q^v_1 q^v_2 + q^v_2q^v_1 = -2(q^v_1 \cdot q^v_2)$<jupyter_code>ipy.display.YouTubeVideo('https://www.youtube.com/watch?v=RbKjaPDoGlw&list=PLpzmRsG7u_gqaTo_vEseQ7U8KFvtiJY4K&index=11',width=800,height=640)<jupyter_output><empty_output>
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