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--- lecture_20_slides [2016/04/13 15:38] – [Example 2] rupert
+++ lecture_20_slides [2016/04/13 15:43] (current) – rupert
@@ Line 172: / Line 172: @@
   * $\left[\begin{smallmatrix}1&-1&2&1\\3&2&-1&4\end{smallmatrix}\right]\to_{EROs}\left[\begin{smallmatrix}1&0&3/5&6/5\\0&1&-7/5&1/5\end{smallmatrix}\right]$
   * Line $L$ of intersection is $\c xyz=\c{6/5}{1/5}0+t\c{-3/5}{7/5}1$, $t\in\mathbb{R}$.
-  * $\c{-\tfrac35}{\tfrac75}1$ is a direction vector along $L$
-  * Another one is $5\c{-\tfrac35}{\tfrac75}1=\c{-3}75$
-  * So $\c{-3}75$ is a vector in the plane $\Pi$.
-  * Take $t=2$ gives $(0,3,2)$ in $L$, so this is in $\Pi$.
 ==== ====
+  * $\Pi$ contains $L:\c xyz=\c{6/5}{1/5}0+t\c{-3/5}{7/5}1$, $t\in\mathbb{R}$, orthogonal to $\Pi_3: 2x+y+z=3$
-Since $\Pi$ is perpendicular to $\Pi_3$, which has normal vector $\nn_3=\c211$, the vector $\c211$ is in $\Pi$.
+  * $5\c{-3/5}{7/5}1=\c{-3}75$ is a vector along $L$, so in $\Pi$
+  * $\Pi_3$ has  normal vector $\nn_3=\c211$, which is in $\Pi$.
-So a normal vector for $\Pi$ is
+  * Normal vector for $\Pi$: $\nn=\c211\times\c{-3}75 = \cp211{-3}75=\c{-2}{-13}{17}$
-\[ \nn=\c211\times\c{-3}75 = \cp211{-3}75=\c{-2}{-13}{17}\]
+  * $\Pi$ has equation $-2x-13y+17z=d$
-hence $\Pi$ has equation $-2x-13y+17z=d$, and subbing in the point $(0,3,2)$ gives $d=0-13(3)+17(2)=-39+34=-5$, so $\Pi$ has equation $-2x-13y+17z=-5$, or
+==== ====
-\[ 2x+13y-17z=5.\]
+  * $\Pi$ has equation $-2x-13y+17z=d$ and contains $L:\c xyz=\c{6/5}{1/5}0+t\c{-3/5}{7/5}1$, $t\in\mathbb{R}$
+  * Take $t=2$: $(0,3,2)$ in $L$, so in $\Pi$
+  * Sub in: $d=0-13(3)+17(2)=-39+34=-5$
+  * Answer: $-2x-13y+17z=-5$, or $2x+13y-17z=5$.