Plugin installed incorrectly. Rename plugin directory '_include' to 'include'.
Plugin installed incorrectly. Rename plugin directory '__include' to 'include'.
lecture_18_slides
Differences
This shows you the differences between two versions of the page.
| Both sides previous revisionPrevious revisionNext revision | Previous revision | ||
| lecture_18_slides [2017/04/05 15:35] – [Example 1] rupert | lecture_18_slides [2017/04/10 16:30] (current) – [Proof of the geometric dot product formula] rupert | ||
|---|---|---|---|
| Line 75: | Line 75: | ||
| * Consider a triangle with two sides $\vv$ and $\ww$. By the triangle rule for vector addition, the third side is $\vv-\ww$: | * Consider a triangle with two sides $\vv$ and $\ww$. By the triangle rule for vector addition, the third side is $\vv-\ww$: | ||
| * Use the cosine rule with $A=\theta$, $a=\|\vv-\ww\|$, | * Use the cosine rule with $A=\theta$, $a=\|\vv-\ww\|$, | ||
| - | * $ \|\vv-\ww\|^2=\|\vv\|\,\|\ww\|-2\|\vv\|\, | + | * $ \|\vv-\ww\|^2=\|\vv\|^2+\|\ww\|^2-2\|\vv\|\, |
| ==== Proof of the geometric dot product formula, continued==== | ==== Proof of the geometric dot product formula, continued==== | ||
| Line 104: | Line 104: | ||
| ==== Corollary 2: orthogonal vectors ==== | ==== Corollary 2: orthogonal vectors ==== | ||
| If $\vv$ and $\ww$ are non-zero vectors with $\vv\cdot\ww=0$, | If $\vv$ and $\ww$ are non-zero vectors with $\vv\cdot\ww=0$, | ||
| + | *Proof: the angle formula gives $\cos \theta=\frac{\vv\cdot\ww}{\|\vv\|\, | ||
| + | *So $\cos\theta=0$ | ||
| + | *So $\theta$ is a right-angle. | ||
| ==== Example 2 ==== | ==== Example 2 ==== | ||
| Prove that $A=(2,3)$, $B=(3,6)$ and $C=(-4,5)$ are the vertices of a right-angled triangle. | Prove that $A=(2,3)$, $B=(3,6)$ and $C=(-4,5)$ are the vertices of a right-angled triangle. | ||
| Line 118: | Line 120: | ||
| * Observe that $\ww=\c{-2}1$ has $\vv\cdot\ww=0$ | * Observe that $\ww=\c{-2}1$ has $\vv\cdot\ww=0$ | ||
| * So $\vv$ and $\ww$ are orthogonal | * So $\vv$ and $\ww$ are orthogonal | ||
| - | * $\vec u=\frac1{\|\ww\|}\ww$ is a unit vector in the same direction as $\ww$, (so is also orthogonal to $\vv$). | + | * $\vec u=\frac1{\|\ww\|}\ww$ is a unit vector in the same direction as $\ww$ |
| + | * so it is also orthogonal to $\vv$. | ||
| * So $\vec u=\frac1{\sqrt5}\c{-2}1=\c{-2/ | * So $\vec u=\frac1{\sqrt5}\c{-2}1=\c{-2/ | ||
| - | ===== Orthogonal projection ===== | ||
| - | |||
| - | 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 $\ww$. | ||
| - | |||
| - | * {{ : | ||
| - | |||
| - | * We write $\pp=\def\ppp{\text{proj}_{\ww}\vv}\ppp$. | ||
| - | * $\nn=\vv-\pp$ is **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 $\pp\cdot\ww=\vv\cdot\ww$ | ||
| - | * 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 ==== | ||
| - | $\vv=\def\c# | ||
| - | * $\ppp=\frac{\vv\cdot\ww}{\|\ww\|^2}\ww$ | ||
| - | * $=\frac{2-2-4}{2^2+(-1)^2+4^2}\c2{-1}4$ | ||
| - | * $=-\frac4{21}\c2{-1}4$ | ||
| - | * component of $\vv$ orthogonal to $\ww$ is $\nn=\vv-\ppp$ | ||
| - | * $\nn=\c12{-1}-\left(-\frac4{21}\right)\c2{-1}4=\frac1{21}\c{29}{38}{-5}$. | ||
lecture_18_slides.1491406544.txt.gz · Last modified: by rupert