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--- lecture_19_slides [2017/04/10 15:47] – [Formula for $\pp=\ppp$] rupert
+++ lecture_19_slides [2017/04/11 09:56] (current) – [Corollary: the length of $\vec v\times\vec w$] rupert
@@ Line 64: / Line 64: @@
   * Let $\def\vc#1{\left[\begin{smallmatrix}#1_1\\#1_2\\#1_3\end{smallmatrix}\right]}\vec v=\vc v$ and $\vec w=\vc w$ be vectors in $\mathbb{R}^3$
   * $\vv\times\ww$ is another vector in $\mathbb{R}^3$, the cross product of $\vv$ and $\ww$
-  * ... defined as the determinant $\vv\times\ww=\def\cp#1#2#3#4#5#6{\left|\begin{smallmatrix}\i&\j&\k\\#1&#2&#3\\#4&#5&#6\end{smallmatrix}\right|}\def\cpc#1#2{\cp{#1_1}{#1_2}{#1_3}{#2_1}{#2_2}{#2_3}}\cpc vw.$
+  * ... defined as the determinant $\color{blue}{\vv\times\ww=\def\cp#1#2#3#4#5#6{\left|\begin{smallmatrix}\i&\j&\k\\#1&#2&#3\\#4&#5&#6\end{smallmatrix}\right|}\def\cpc#1#2{\cp{#1_1}{#1_2}{#1_3}{#2_1}{#2_2}{#2_3}}\cpc vw.}$
   * Interpret this by Laplace expansion along row 1:\[\cpc vw=\def\vm#1{\left|\begin{smallmatrix}#1\end{smallmatrix}\right|}\vm{v_2&v_3\\w_2&w_3}\i-\vm{v_1&v_3\\w_1&w_3}\j+\vm{v_1&v_2\\w_1&w_2}\k=\c{v_2w_3-v_3w_2}{-(v_1w_3-v_3w_1)}{v_1w_2-v_2w_1}\]
 ==== Example ====
@@ Line 96: / Line 97: @@
   * Calculations/properties of the determinant.
-==== Theorem ====
+==== Theorem: cross and dot product formula ====
 For any vectors $\vv$ and $\ww$ in $\mathbb{R}^3$, we have
 \[ \|\vv\times\ww\|^2+(\vv\cdot\ww)^2=\|\vv\|^2\,\|\ww\|^2.\]
@@ Line 106: / Line 107: @@
 For any vectors $\vv$ and $\ww$ in $\mathbb{R}^3$, we have
 \[ \|\vv\times\ww\|=\|\vv\|\,\|\ww\|\,\sin\theta\]
-where $\theta$ is the angle between $\vv$ and $\ww$ (with $0\le\theta<\pi$).
+where $\theta$ is the angle between $\vv$ and $\ww$ (with $0\le\theta\le\pi$).
 === Proof ===
-  * We know that $\vv\cdot\ww=\|\vv\|\,\|\ww\|\cos\theta$.
+  * Geometric dot product formula: $\vv\cdot\ww=\|\vv\|\,\|\ww\|\cos\theta$
+  * $\times$ & $\cdot$ formula: $\|\vv\times\ww\|^2+(v\cdot w)^2=\|\vv\|^2\,\|\ww\|^2$
   * So $\|\vv\times\ww\|^2=\|\vv\|^2\,\|\ww\|^2-(\vv\cdot\ww)^2$
     * $=\|\vv\|^2\,\|\ww\|^2-\|\vv\|^2\|\ww\|^2\cos^2\theta$
     * $=\|\vv\|^2\,\|\ww\|^2(1-\cos^2\theta)$
     * $=\|\vv\|^2\,\|\ww\|^2\sin^2\theta$.
-  * $\sin\theta\ge0$ for $0\le\theta<\pi$, so taking square roots of both sides gives $ \|\vv\times\ww\|=\|\vv\|\,\|\ww\|\,\sin\theta$. ■
+  * $\sin\theta\ge0$ for $0\le\theta\le\pi$, so taking square roots of both sides gives $ \|\vv\times\ww\|=\|\vv\|\,\|\ww\|\,\sin\theta$. ■
 ===== Geometry of the cross product =====