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\centerline
{
\bf Solutions to the ``QUIZ'' for Sept. 8, 2009
}
\bigskip
{\bf 1.} Determine whether the two vectors are orthogonal
and if not, whether the angle  between them is acute or obtuse.
${\bf a.} \quad \langle 1,1,1 \rangle \quad , \quad \langle 3,-2,-1 \rangle \quad .$

${\bf b.} \quad \quad \langle 4,3 \rangle \quad , \quad  \langle 2,-4 \rangle \quad .$

{\bf Solution}: {\bf a.}
$$
\quad \langle 1,1,1 \rangle \cdot \langle 3,-2,-1 \rangle=
1 \cdot 3 + 1 \cdot (-2) + 1 \cdot (-1)= 3-2-1=0 \quad.
$$
{\bf Ans.}: Since the dot-product is $0$, the vectors are perpendicular.

{\bf b.}
$$
\langle 4,3 \rangle \cdot \langle 2,-4 \rangle=
4 \cdot 2 + 3 \cdot (-4)=8 -12 =-4 \quad .
$$
{\bf Ans.}: Since the dot-product is {\bf not}
$0$, the vectors are {\bf not} perpendicular.
Since the dot-product is {\bf negative}, the angle is {\bf obtuse}.

{\bf Comments:} Most people got it right. Quite a few people,
for {\bf b}, volunteered more information, actually finding the
angle. This is nice, but not necessary! Answer what you have been
asked, not less, but also not more!


{\bf 2.}  Calculate ${\bf v} \times {\bf w}$, \quad if
$$
{\bf v}= \langle 0,1, -1 \rangle \quad , \quad
{\bf w}= \langle 1,-1, 0 \rangle \quad .
$$

$$
v \times w=
$$
$$
\left \vert \matrix{ {\bf i} & {\bf j} & {\bf k}  \cr
0 & 1& -1\cr
1 & -1 & 0 
} \right \vert \quad=
$$
$$
{\bf i} \left \vert \matrix{ 
1 & -1\cr
-1 & 0 
} \right \vert 
-{\bf j} \left \vert \matrix{ 
0 & -1\cr
1 & 0 
} \right \vert 
+{\bf k} \left \vert \matrix{ 
0 & 1 \cr
1 & -1 
} \right \vert 
$$
$$
={\bf i} (1 \cdot (0)-(-1) \cdot (-1))
-{\bf j} (0 \cdot (0)-(-1) \cdot (1))
+{\bf k} (0 \cdot (-1)-(1) \cdot (1))=
$$
$$
-{\bf i}-{\bf j}-{\bf k} \quad.
$$
{\bf Finally} converting to the {\bf usual} notation, we get
$\langle -1,-1,-1 \rangle$.

{\bf Ans.:} $\langle -1,-1,-1 \rangle$.

{\bf Comments}: About \%80 of the people got it perfectly.
Another \%10 set-it up correctly, but messed up the arithmetic
(be careful with the minus signs!:
minus times ninus is plus,
minus times minus times minus is minus, etc.

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