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--- lecture_23_slides [2017/04/24 11:06] – [Skew lines in $\mathbb{R}^3$] rupert
+++ lecture_23_slides [2017/04/24 11:30] (current) – [Example using $\def\dist{\text{dist}}\dist(B,L)=\frac{\|\vec{AB}\times\vv\|}{\|\vv\|}$] rupert
@@ Line 23: / Line 23: @@
-==== Example using $\def\dist{\text{dist}}\dist(B,L)=\frac{\|\vec{AB}\times\vv\|}{\|\vv\|}$ ====
+==== Example using $\def\vv{\vec v}\def\dist{\text{dist}}\dist(B,L)=\frac{\|\vec{AB}\times\vv\|}{\|\vv\|}$ ====
 Find the distance from the point $B=(1,2,3)$ to the line \[L:\c xyz=\c10{-1}+t\c41{-5},\quad t\in\mathbb{R}.\]
@@ Line 49: / Line 49: @@
   * $\dist(B,L)=\|\nn\|=\frac17\sqrt{12^2+17^2+13^2} \approx 3.5051$.
-===== Distance between skew lines in $\mathbb{R}^3$ =====
+===== Skew lines in $\mathbb{R}^3$ =====
   * Two lines are skew if they are not parallel, and do not cross.
-  * Take two skew lines, $L_1$ and $L_2$
+  * Skew lines only exist in $\bR^n$ for $n\ge3$.
+  * Take skew lines $L_1$ and $L_2$ in $\rt$, direction vectors $\vv_1,\vv_2${{ :sl1.png?nolink&400 |}}
   * Let's find a formula for $\dist(L_1,L_2)$
-  * $\vv_1$, $\vv_2$: direction vectors {{ :sl1.png?nolink&400 |}}
   * $\dist(L_1,L_2)$ is measured orthogonal to both $\vv_1$ and $\vv_2$
   * i.e. in direction of $\nn=\vv_1\times\vv_2$.
 ==== ====
-  * Let $\Pi$ = plane with normal vector $\nn$ which contains $L_1$.{{ :sl2.png?nolink&600 |}}
+  * Let $\Pi$ = plane with normal vector $\nn$ which contains $L_1$.{{ :sl2.png?nolink&800 |}}
   * Let $B$ be in $L_2$, and $A$ in $\Pi$ (e.g. any $A$ in $L_1$)
   * Then $\dist(L_1,L_2)=\dist(B,\Pi) = \frac{|\vec{AB}\cdot \nn|}{\|\nn\|}$
-  * (This actually works for any non-parallel lines, even if they cross)
 ==== Example ====
@@ Line 73: / Line 72: @@
   * $\vec{AB}=\c 220$
   * So $\dist(L_1,L_2)=\frac{|\vec{AB}\cdot \nn|}{\|\nn\|}=\frac{2(5)+2(2)+0(-3)}{\sqrt{5^2+2^2+3^2}} = \frac{14}{\sqrt{38}}$.
+==== 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)
+    * actually: any non-parallel lines $L_1$, $L_2$
+  * 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 $\dist(L_1,L_2)=\dist(A,L_2)$ for any point $A$ in $L_1$
+    * So we can use one of of the point-to-line distance formulae we saw earlier.
+==== End of the course ====
+  * Next time: some examples from the most recent exam paper
+  * Have a look at these before Thursday's lecture to prepare