Differences

This shows you the differences between two versions of the page.

--- lecture_19_slides [2016/04/11 15:16] – [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 $\vec v\cdot \vec w=0$
-  * Orthogonal projection of $\vec v$ onto $\vec w$ is $\vec p=\frac{\vec v\cdot \vec w}{\|\vec w\|^2}\vec w$ {{ :t3.png?nolink&300 |}}
-  * $\vec p$ is also called the component of $\vec v$ along $\vec w$
+===== Orthogonal projection =====
-  * $\vec n=\vec v-\vec p$ is call the component of $\vec v$ orthogonal to $\vec w$
+Let $\def\pp{\vec p}\def\ww{\vec w}\def\vv{\vec v}\def\nn{\vec n}\ww$ non-zero, and $\vv$ any vector.
+$\pp$ is the **orthogonal projection of $\vv$ onto $\ww$** if:
+  - $\pp$ is in the same direction as $\ww$; and
+  - the vector $\nn=\vv-\pp$ is orthogonal to $\ww$.
+  * {{ :t3.png?nolink&400 |}}
+  * We write $\pp=\def\ppp{\text{proj}_{\ww}\vv}\ppp$.
+  * $\nn=\vv-\pp$ is called **the component of $\vv$ orthogonal to $\ww$**.
+==== Formula for $\pp=\ppp$ ====
+  * $\nn=\vv-\pp$ 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 $c\|\ww\|^2=\vv\cdot\ww$
+    * so $c=\frac{\vv\cdot\ww}{\|\ww\|^2}$.
+  * So $\color{blue}{\pp=\ppp=\frac{\vv\cdot\ww}{\|\ww\|^2}\ww}.$
 ==== Example ====
-$\def\vv{\vec v}\def\pp{\vec p}\def\ppp{\text{proj}_{\ww}\vv}\def\ww{\vec w}\def\nn{\vec n}\vv=\def\c#1#2#3{\left[\begin{smallmatrix}#1\\#2\\#3\end{smallmatrix}\right]}\c12{-1}$ and $\ww=\c2{-1}4$
+$\vv=\def\c#1#2#3{\left[\begin{smallmatrix}#1\\#2\\#3\end{smallmatrix}\right]}\c12{-1}$ and $\ww=\c2{-1}4$
   * $\ppp=\frac{\vv\cdot\ww}{\|\ww\|^2}\ww$
     * $=\frac{2-2-4}{2^2+(-1)^2+4^2}\c2{-1}4$
     * $=-\frac4{21}\c2{-1}4$
   * component of $\vv$ orthogonal to $\ww$ is $\nn=\vv-\ppp$
-    * $\nn=\c12{-1}-\left(-\frac4{21}\right)\c2{-1}4=\c{29/21}{38/21}{-5/21}$.
+    * $\nn=\c12{-1}-\left(-\frac4{21}\right)\c2{-1}4=\frac1{21}\c{29}{38}{-5}$.
 ===== The cross product of vectors in $\mathbb{R}^3$ =====
@@ Line 34: / 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 66: / 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 76: / 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 =====