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--- lecture_23 [2017/04/20 08:58] – rupert
+++ lecture_23 [2017/05/06 09:59] (current) – rupert
@@ Line 1: / Line 1: @@
 ===== 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 $L$.
@@ Line 17: / Line 17: @@
 To find the distance from the point $B=(1,2,3)$ to the line \[L:\def\c#1#2#3{\begin{bmatrix}#1\\#2\\#3\end{bmatrix}}\c xyz=\c10{-1}+t\c41{-5},\quad t\in\mathbb{R}\]
-we can choose $A=\c10{-1}$ so that $\vec{AB}=\c024$ and taking $\vv=\c41{-5}$, we obtain
+we can choose $A=(1,0,{-1})$ so that $\vec{AB}=\c024$ and taking $\vv=\c41{-5}$, we obtain
 \[ \vec{AB}\times \vv = \begin{vmatrix}\vec\imath&\vec\jmath&\vec k\\0&2&4\\4&1&-5\end{vmatrix}=\c{-14}{16}{-8}=2\c{-7}{8}{-4}\]
 so
 \[ \def\dist{\text{dist}}\dist(B,L)=\frac{\|\vec{AB}\times\vv\|}{\|\vv\|}=\frac{2\sqrt{7^2+8^2+4^2}}{\sqrt{4^2+1^2+5^2}} = \frac{2\sqrt{129}}{\sqrt{42}} \approx 3.5051.\]
-==== Cross product method (only for $\rt$) ====
+==== 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 $n$.
@@ Line 41: / Line 41: @@
 \[ \dist(B,L)=\|\nn\|=\frac17\sqrt{12^2+17^2+13^2} = \frac17\sqrt{602} \approx 3.5051.\]
-===== 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.
@@ Line 85: / Line 87: @@
 and if $A=(1,0,1)$ and $B=(3,2,1)$ then $A$ and $B$ are points with one in $L_1$ and the other in $L_2$, and $\vec{AB}=\c 220$. Hence
 \[\dist(L_1,L_2)=\frac{|\vec{AB}\cdot \nn|}{\|\nn\|}=\frac{2(5)+2(2)+0(-3)}{\sqrt{5^2+2^2+3^2}} = \frac{14}{\sqrt{38}}.\]
+==== Distance between lines in $\mathbb{R}^3$ in general ====
+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), as we saw above,
+  * 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?
+    * The formula can't work because we'd have $\vec v_1=\vec v_2$ so $\vec n=\vec v_1\times \vec v_2=\vec 0$
+    * Instead: observe that when $L_1$ and $L_2$ are parallel, we have $\dist(L_1,L_2)=\dist(A,L_2)$ for any point $A$ in $L_1$
+    * So we can use one of of the point-to-line distance formulae we saw earlier.