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%%a Plain TeX file by Doron Zeilberger for Math 251(1 page)

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\centerline
{
\bf Dr. Z's Math251 Handout \#12.5 [Equations of Lines and Planes]
}

By Doron Zeilberger

{\bf Problem Type 12.5a}: 
Find an equation of the plane that passes through three given points
%%General Problem

{\bf Example Problem 12.5a}: 
Find an equation of the plane that passes through the points
$(1,1,1)$, $(2,0,1)$, $(2,1,0)$.


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\halign{           
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{\bf Steps} & {\bf Example} \cr  &  \cr
{\bf 1.} Calling these points $P,Q,R$ find the
vectors ${\bf PQ}$ and ${\bf PR }$ by
doing $Q-P$ and $R-P$.
&
{\bf 1.} 
$P=(1,1,1)$, $Q=(2,0,1)$, $R=(2,1,0)$,
$$
{\bf PQ}=Q-P=\langle 2-1,0-1,1-1 \rangle = \langle 1,-1,0 \rangle \quad ,
$$
$$
{\bf PR}=R-P=\langle 2-1,1-1,0-1 \rangle = \langle 1,0,-1 \rangle \quad .
$$

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{\bf 2.} 
Find a vector normal to the plane by computing the cross-product
${\bf PQ} \times {\bf PR}$ \quad .
%%% End General Step 2
&
{\bf 2.} 
$$
{\bf PQ} \times {\bf PR}=
\langle 1,-1,0 \rangle \times \langle 1,0,-1 \rangle
=
$$

$$
\left \vert \matrix{ {\bf i} & {\bf j} & {\bf k}  \cr
1 & -1& 0\cr
1 & 0 & -1 
} \right \vert \quad=
$$
$$
{\bf i} \left \vert \matrix{ 
-1 & 0\cr
0 & -1 
} \right \vert 
-{\bf j} \left \vert \matrix{ 
1 & 0\cr
1 & -1 
} \right \vert 
+{\bf k} \left \vert \matrix{ 
1 & -1 \cr
1 & 0 
} \right \vert 
$$
$$
={\bf i}+{\bf j}+{\bf k}
=\langle 1,1,1 \rangle \quad.
$$

This is the {\bf normal vector}
${\bf n}=\langle a,b,c \rangle $. So 
${\bf n}=\langle a,b,c \rangle=\langle 1,1,1 \rangle$
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{\bf 3.} Pick any of the three points (it does not matter which)
as the refence point $(x_0,y_0,z_0)$ and use the formula
$$
a(x-x_0)+b(y-y_0)+c(z-z_0)=0 \quad .
$$
%%% General Step 3
&
{\bf 3.} 
Picking $P$ we get $(x_0,y_0,z_0)=(1,1,1)$, since
$a=1,b=1,c=1$, we get that an equation is
$$
1 \cdot (x-1)+1 \cdot (y-1)+1 \cdot (z-1)=0 \quad
$$
and expanding we get
$$
x+y+z=3 \quad.
$$

{\bf Ans.:} An equation for the plane passing through 
$P,Q$ and $R$ is $x+y+z=3$.

{\bf Check:} Plug-in all the three points into the equation
and make sure that they agree.

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{\bf Problem Type 12.5b}: 
Find direction numbers for the line of intersections of the planes
$a_1x+b_1y+c_1z=d_1$ and
$a_2x+b_2y+c_2z=d_2$.
%%General Problem

{\bf Example Problem 12.5b}: 
Find direction numbers for the line of intersections of the planes
$2x+3y+4z=2$ and
$-3x+2y+3z=1$.

%%Specific Problem


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{\bf Steps} & {\bf Example} \cr  &  \cr
{\bf 1.} By looking at the coeffs. of $x,y,z$ extract the
{\bf normal vectors}
${\bf n_1}=\langle a_1,b_1,c_1 \rangle$ and
${\bf n_2}=\langle a_2,b_2,c_2 \rangle$.

{\bf Note:} the numbers on the right sides ($d_1,d_2$) are not needed.
&
{\bf 1.} ${\bf n_1}=\langle 2,3,4 \rangle$ and
${\bf n_2}=\langle -3,2,3 \rangle$.
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{\bf 2.} Take the cross-product ${\bf n_1} \times {\bf n_2}$.
The components are the {\bf direction numbers} of the
line of intersection.
&
{\bf 2.} 
$$
{\bf n_1} \times {\bf n_2}=
\langle 2,3,4 \rangle \times \langle -3,2,3 \rangle =
\langle 1,-18,13 \rangle \quad .
$$
(You do it!)

{\bf Ans.}: The direction numbers are $\langle 1,-18,13 \rangle$.
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\end
