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\centerline{\bf MAT 205  SPRING 2000}


\centerline{\bf MIDTERM I}


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NAME (first and last; readable): 
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SSN: 

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\centerline{\bf THERE ARE FOUR PROBLEMS PLUS ONE EXTRA CREDIT PROBLEM}

\centerline{\bf SHOW YOUR WORK!!!}

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{\bf 1)}
Let
$$
{\bf a} =<2,1,-1>, \quad {\bf b} = < 1,-2,3 >, \quad {\bf c} = < 1, 0, -2 >. 
$$

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A) Compute $   {\bf b} \cdot ({\bf a} + {\bf c}).$



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B) Let ${\bf d}= <2, m, 1>$ be a vector varying with  the parameter
$m$. Find the values of $m$, if any,
that make ${\bf d}$ and ${\bf a}$ perpendicular.

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NAME (first and last; readable): 
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SSN: 

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{\bf 2)} Let 
$$
P= (-1,1,-1), \qquad Q= (1,-1,2), \qquad R=(4,0,3)
$$
be three points in space.

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Find the plane passing through $P$, $Q$ and $R$.

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NAME (first and last; readable): 
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SSN: 

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{\bf 3)} Sketch the  surface
$$
y= \sqrt{x^2 + z^2}.
$$
Give also a brief explanation {\it in words} as to why it looks like you
have drawn it.

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NAME (first and last; readable): 
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SSN: 

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{\bf 4)} A particle is moving in space with position vector ${\bf r}(t)$.

The acceleration is
$$
{\bf a}(t) = {\bf i} - 2 {\bf k}.
$$
The velocity at time $t=0$ is 
$$
{\bf v} (0)= {\bf i} - {\bf j} + {\bf k}.
$$
The position at time $t=0$ is 
$$
{\bf r} (0)= 2{\bf i} - 2{ \bf k}.
$$
Find the position vector ${\bf r} (2)$ at time $t=2$.

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NAME (first and last; readable): 
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SSN: 

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{\bf THIS IS FOR EXTRA CREDIT. ATTEMPT THE FIRST FOUR PROBLEMS FIRST.}


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{\bf 5)} Let ${\bf a} $ and ${\bf b}$ be non zero vectors.

Show that the vector 
$$
 {\bf b} -  proj_{{\bf a}} {\bf b} 
$$
is orthogonal to the vector ${\bf a}$.



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