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--- lecture_19_slides [2017/04/06 10:15] – [Example] rupert
+++ lecture_19_slides [2017/04/11 09:56] (current) – [Corollary: the length of $\vec v\times\vec w$] rupert
@@ Line 5: / Line 5: @@
   * Dot product: $\vec v\cdot\vec w=v_1w_1+v_2w_2+\dots + v_nw_n$
   * Geometric formula: $\vec v\cdot \vec w = \|\vec v\|\,\|\vec w\|\,\cos\theta$
+    * $\|\vec v\|=$length of $\vec v$ $=\sqrt{v_1^2+\dots+v_n^2}$
+    * $\|\vec w\|=$length of $\vec w$ $=\sqrt{w_1^2+\dots+w_n^2}$
+    * $\theta=$angle between $\vec v$ and $\vec w$
   * $\vec v$ and $\vec w$ are orthogonal (at right-angles) if $\vec v\cdot \vec w=0$
@@ Line 19: / Line 22: @@
   * We write $\pp=\def\ppp{\text{proj}_{\ww}\vv}\ppp$.
-  * $\nn=\vv-\pp$ is **the component of $\vv$ orthogonal to $\ww$**.
+  * $\nn=\vv-\pp$ is called **the component of $\vv$ orthogonal to $\ww$**.
 ==== Formula for $\pp=\ppp$ ====
-  * $\pp$ same direction as $\ww$, so $\pp=c\ww$, some scalar $c$
   * $\nn=\vv-\pp$ is orthogonal to $\ww$, so $\nn\cdot \ww=0$.
     * so $(\vv-\pp)\cdot \ww=0$
     * so $\vv\cdot\ww-\pp\cdot\ww=0$
     * so $\pp\cdot\ww=\vv\cdot\ww$
+  * $\pp$ in same direction as $\ww$, so $\color{blue}{\pp=c\ww}$ for some scalar $c$
     * so $c\ww\cdot \ww=\vv\cdot\ww$
     * so $c\|\ww\|^2=\vv\cdot\ww$
@@ Line 61: / 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 93: / 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 103: / 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 =====