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\centerline
{
\bf Dr. Z's Math251 Handout (2nd ed.) \#12.2 [Vectors in three dimensions]
}

By Doron Zeilberger

{\bf Problem Type 12.2a}: 
Show that the triangle with vertices
$P=(p_1,p_2,p_3)$, $Q=(q_1,q_2,q_3)$, $R=(r_1,r_2,r_3)$
is an equilateral triangle.

{\bf Example Problem 12.2a}: 
Show that the triangle with vertices
$P=(-4,8,0)$, $Q=(2,4,-2)$, $R=(-2,2,4)$
is an equilateral triangle.

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{\bf Steps} & {\bf Example} \cr  &  \cr
{\bf 1.} 
%%%General Step 1
Use the distance formula
$$
\vert P_1P_2 \vert=
$$
$$
\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2} \quad,
$$
for the distance between two points $P_1(x_1,y_1,z_1)$ and
$P_2(x_2,y_2,z_2)$, to find the three distances
$\vert PQ \vert$, $\vert PR \vert$, $\vert QR \vert$.



%%%End General Step 1
&
{\bf 1.} Here $P=(-4,8,0)$, $Q=(2,4,-2)$, $R=(-2,2,4)$, so
$$
\vert PQ \vert=\sqrt{(2-(-4))^2+(4-8)^2+((-2)-0)^2}=
\sqrt{36+16+4}
$$
$$
=\sqrt{56} \quad.
$$
$$
\vert PR \vert=\sqrt{((-2)-4)^2+(2-8)^2+(4-0)^2}=
\sqrt{4+36+16}
$$
$$
=\sqrt{56} \quad.
$$
$$
\vert QR \vert=\sqrt{((-2)-2)^2+(2-4)^2+(-4-(-2))^2}=\sqrt{16+4+36}
$$
$$
=\sqrt{56} \quad.
$$

%%%End Specific Step 1
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{\bf 2.} 
%%% General Step 2
Check whether theese three distances are all the  same.
It there are, it is an equilateral triangle, otherwise not.
%%% End General Step 2
&
{\bf 2.} 
%%%Specific Step 2
All the distances are the same ($\sqrt{56}$).
{\bf Ans.}: It is an equilateral triangle since
all the  sides have equal length , namely: $\sqrt{56}$.
%%%End Specific Step 2
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{\bf Problem Type 12.2b}: Find an equation of the sphere with center
$C(h,k,l)$ and radius $r$.

{\bf Example Problem 12.2b}: Find an equation of the sphere with center
$(1,2,-1)$ and radius $2$.
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{\bf Steps} & {\bf Example} \cr  &  \cr
{\bf 1.} Implement the formula
$$
(x-h)^2+(y-k)^2+(z-l)^2=r^2
$$
&
{\bf 1.} In this problem $(h,k,l)=(1,2,-1)$ and $r=2$ so the
equation is:
$$
(x-1)^2+(y-2)^2+(z-(-1))^2=2^2 \quad,
$$
which is the same as
$$
(x-1)^2+(y-2)^2+(z+1)^2=2^2 \quad .
$$

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{\bf 2.} Expand everything and move everything to the left
leaving $0$ at the right side. Also rearange terms so that
the quadratic terms come before the linear terms.
&
{\bf 2.} 
$$
x^2-2x+1+y^2-4y+4+z^2+2z+1=4 \quad,
$$
Cleaning up:
$$
x^2+y^2+z^2-2x-4y+2z+2=0 \quad .
$$

{\bf Ans.}: $x^2+y^2+z^2-2x-4y+2z+2=0$.
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}

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{\bf Problem Type 12.2c}: Show that the equation represents a sphere,
and find the center and radius.
$$
x^2+y^2+z^2+ax+by+cz+d=0 \quad .
$$

{\bf Example Problem 12.2c}: Show that the equation represents a sphere,
and find the center and radius.
$$
x^2+y^2+z^2-2x-4y+2z+2=0 \quad .
$$
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{\bf Steps} & {\bf Example} \cr  &  \cr
{\bf 1.} The coefficients of $x^2,y^2,z^2$ should all be
the same! If they are not, for example, if the
equation is $x^2+y^2+3z^2+ 2x+6y-5+11=0$ where the
coefficients are not all the same, then it is {\bf not}
a sphere.
Usually they are all $1$,
If the coefficient of $x^2$ (=coeff. of $y^2$=coeff. of $z^2$) is not
$1$, divide the whole equation by that coefficient, making it $1$. The
coeffs. of $x^2$, $y^2$, $z^2$ should now be {\it all} $1$.
Now group the terms so that $x^2$ is next to the $x$ term,
$y^2$ is next to the $y$ term,
and $z^2$ is next to the $z$ term.
&
{\bf 1.} In this problem, the coeffs. of $x^2$ is already $1$,
as are those of $y^2$ and $z^2$. Grouping the $x$-terms, 
$y$-terms and $z$-terms, we get:
$$
x^2-2x+ y^2-4y+ z^2+2z+2=0
$$

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{\bf 2.} For each part separately, {\it complete the square},
using $X^2+aX=(X+a/2)^2-(a/2)^2$
&
{\bf 2.} 
$$
x^2-2x=(x-1)^2-1
$$
$$
y^2-4y=(y-2)^2-4
$$
$$
z^2+2z=(z+1)^2-1
$$
So 
$$
x^2-2x+ y^2-4y+ z^2+2z+2=0
$$
becomes
$$
(x-1)^2-1+(y-2)^2-4+(z+1)^2-1+2=0
$$
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{\bf 3.} Move all the numbers to the right and express the
resulting number on the right as $r^2$. Compare with the
equation of the sphere 
$$
(x-h)^2+(y-k)^2+(z-l)^2=r^2 \quad,
$$
and read-off the {\bf center} $(h,k,l)$ and the {\bf radious}, $r$.

&
{\bf 3.} 
$$
(x-1)^2-1+(y-2)^2-4+(z+1)^2-1+2=0
$$
is the same as
$$
(x-1)^2+(y-2)^2+(z+1)^2=4
$$
which, in turn, is the same as
$$
(x-1)^2+(y-2)^2+(z-(-1))^2=2^2 \quad,
$$
which is an equation of a sphere with {\bf center} $(1,2,-1)$ and
{\bf radius} $2$.

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