Added: cross-product

Author: fangjunzhou

_posts/Quaternions/2022-04-25-cross-product.md

@@ -0,0 +1,83 @@
+---
+layout: post
+title: Fucking Cross Product
+category: quaternions
+---
+
+I'm fucked up with cross product. So I decide to write this shit.
+
+# Defination
+
+The basic defination of cross product comes from matrix determinant. Write i, j, and k in the first row and two corresponding vector in second and third row.
+
+$$
+\begin{aligned}
+    \textbf{a} \times \textbf{b} &= 
+    \begin{vmatrix}
+        i & j & k \\
+        a_x & a_y & a_z \\
+        b_x & b_y & b_z
+    \end{vmatrix} \\
+    &=
+    \begin{vmatrix}
+        a_y & a_z \\
+        b_y & b_z
+    \end{vmatrix} i
+    +
+    \begin{vmatrix}
+        a_x & a_z \\
+        b_x & b_z
+    \end{vmatrix} j
+    +
+    \begin{vmatrix}
+        a_x & a_y \\
+        b_x & b_y
+    \end{vmatrix} k \\
+    &= (a_yb_z - a_zb_y)i + (a_xb_z - a_zb_x)j + (a_xb_y - a_yb_x)k
+\end{aligned}
+$$
+
+From this defination, it's obvious that order matters in cross product.
+
+$$
+\begin{aligned}
+    \textbf{b} \times \textbf{a} &= 
+    \begin{vmatrix}
+        i & j & k \\
+        b_x & b_y & b_z \\
+        a_x & a_y & a_z
+    \end{vmatrix} \\
+    &=
+    \begin{vmatrix}
+        b_y & b_z \\
+        a_y & a_z
+    \end{vmatrix} i
+    +
+    \begin{vmatrix}
+        b_x & b_z \\
+        a_x & a_z
+    \end{vmatrix} j
+    +
+    \begin{vmatrix}
+        b_x & b_y \\
+        a_x & a_y
+    \end{vmatrix} k \\
+    &= (b_ya_z - b_za_y)i + (b_xa_z - b_za_x)j + (b_xa_y - b_ya_x)k \\
+    &= (a_zb_y - a_yb_z)i + (a_zb_x - a_xb_z)j + (a_yb_x - a_xb_y)k \\
+    &= -\textbf{a} \times \textbf{b}
+\end{aligned}
+$$
+
+# Magnitude
+
+$$
+\vert \textbf{b} \times \textbf{a} \vert = \vert \textbf{a} \vert \vert \textbf{b} \vert sin\theta
+$$
+
+# Direction
+
+The **right hand rule** cross product use is:
+
+![picture 1](/Blog/images/2022-04-26-16-10-59-cross-product.png)  
+
+As shown in the picture, use right hand to wrap from a to b, thumb points to the direction of $\textbf{b} \times \textbf{a}$