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lecture_23

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lecture_23 [2017/04/25 09:18] – rupertlecture_23 [2017/05/06 09:59] (current) – rupert
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 ===== The distance from a point to a line ===== ===== The distance from a point to a line =====
  
-==== Dot product method ====+==== Cross product method ====
 Suppose $L$ is a line in $\def\rt{\mathbb{R}^3}\def\rn{\mathbb{R}^n}\rt$. Let $A$ be a point on $L$ and let $\def\vv{\vec v}\vv$ be a direction vector along $L$. Suppose $L$ is a line in $\def\rt{\mathbb{R}^3}\def\rn{\mathbb{R}^n}\rt$. Let $A$ be a point on $L$ and let $\def\vv{\vec v}\vv$ be a direction vector along $L$.
  
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 \[ \def\dist{\text{dist}}\dist(B,L)=\frac{\|\vec{AB}\times\vv\|}{\|\vv\|}=\frac{2\sqrt{7^2+8^2+4^2}}{\sqrt{4^2+1^2+5^2}} = \frac{2\sqrt{129}}{\sqrt{42}} \approx 3.5051.\] \[ \def\dist{\text{dist}}\dist(B,L)=\frac{\|\vec{AB}\times\vv\|}{\|\vv\|}=\frac{2\sqrt{7^2+8^2+4^2}}{\sqrt{4^2+1^2+5^2}} = \frac{2\sqrt{129}}{\sqrt{42}} \approx 3.5051.\]
  
-==== Cross product method ====+==== Dot product method ====
  
 The method above relies on the cross product, so only works in $\def\c#1#2#3{\begin{bmatrix}#1\\#2\\#3\end{bmatrix}}\def\rt{\mathbb{R}^3}\def\rn{\mathbb{R}^n}\def\vv{\vec v}\def\dist{\text{dist}}\rt$. The following alternative method works in $\rn$ for any $n$. The method above relies on the cross product, so only works in $\def\c#1#2#3{\begin{bmatrix}#1\\#2\\#3\end{bmatrix}}\def\rt{\mathbb{R}^3}\def\rn{\mathbb{R}^n}\def\vv{\vec v}\def\dist{\text{dist}}\rt$. The following alternative method works in $\rn$ for any $n$.
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 \[ \dist(B,L)=\|\nn\|=\frac17\sqrt{12^2+17^2+13^2} = \frac17\sqrt{602} \approx 3.5051.\] \[ \dist(B,L)=\|\nn\|=\frac17\sqrt{12^2+17^2+13^2} = \frac17\sqrt{602} \approx 3.5051.\]
  
-===== The distance between skew lines in $\mathbb{R}^3$ =====+===== The distance between lines in $\mathbb{R}^3$ ===== 
 + 
 +==== Skew lines ====
  
 Suppose that $L_1$ and $L_2$ are skew lines in $\rt$: lines which are not parallel and do not cross. Suppose that $L_1$ and $L_2$ are skew lines in $\rt$: lines which are not parallel and do not cross.
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 The formula $\dist(L_1,L_2)=\frac{|\vec{AB}\cdot \nn|}{\|\nn\|}$ where $\nn=\vec v_1\times \vec v_2$ works for The formula $\dist(L_1,L_2)=\frac{|\vec{AB}\cdot \nn|}{\|\nn\|}$ where $\nn=\vec v_1\times \vec v_2$ works for
-  * skew lines (not parallel, not intersecting) +  * skew lines (not parallel, not intersecting), as we saw above, 
-  * actually: any non-parallel lines $L_1$, $L_2$+  * and actually: any non-parallel lines $L_1$, $L_2$. We can see this by noticing that if the lines above intersect, then $L_1$ lies in $\Pi$, so the formula gives $\dist(L_1,L_2)=\dist(B,\Pi)=0$ which is the correct answer.
 What about parallel lines? What about parallel lines?
     * The formula can't work because we'd have $\vec v_1=\vec v_2$ so $\vec n=\vec v_1\times \vec v_2=\vec 0$     * The formula can't work because we'd have $\vec v_1=\vec v_2$ so $\vec n=\vec v_1\times \vec v_2=\vec 0$
lecture_23.1493111914.txt.gz · Last modified: by rupert
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