\begin{problem}{MATH 294}{UNKNOWN}{FINAL}{2}{}
\prob{Let $R(t) = e^t \cos t \vi + t \vj + t e^t \vk$ be position
of a particle moving in space at time $t$. } \probpart{Set up,
but do not evaluate, a definite integral equal to the distance
traveled by the particle from $t = 0$ to $t = \pi.$}
\probpart{Find all points on the curve where the velocity vector
is orthogonal to the acceleration vector.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{UNKNOWN}{\pr{2}}{2}{}
\prob{If $\vec{r}(t) = \cos t \vi - 3 \sin t \vj$ gives the
position of a particle} \probpart{find the velocity and
acceleration} \probpart{sketch the curve, and sketch the
acceleration and velocity vectors at one point of the curve (you
choose the point)} \probpart{what is the torsion (if you can do
this without computation, that is acceptable - but please give
reasons for your answer).}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{UNKNOWN}{FINAL}{1}{}
\probb{Find an equation for the plane containing the points
$(1,0,1),(-1,2,0), (1,1,1).$} \probpart{Find the cosine of the
angle between the plane in a) and the plane $x - 2 y + z - 5 =
0$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{84}}{FINAL}{1}{}
\prob{Prove that for any vector $\vec{F}:$ \[ \vec{F} =
\brc{\vec{F} \cdot \vi}\vi + \brc{\vec{F} \cdot \vj}\vj  +
\brc{\vec{F} \cdot \vk}\vk.
\]}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{85}}{FINAL}{1}{}
\prob{Find a unit vector in $\Re^3$ which is perpendicular to
both $\vi + \vj$ and $\vk$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{85}}{FINAL}{5}{}
\prob{Find a solution defined in the right half-plane $\set{(x,y)
| x > 0 }$ whose  gradient is the vector field
$\frac{-y}{x^2+y^2} \vi + \frac{x}{x^2+y^2} \vj$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{85}}{FINAL}{7}{}
\prob{Let $S$ be the surface with equation $x^2 + x y = z^2 + 2
y.$} \probpart{Find the equation of the plane tangent to $S$ at
the point $(1,0,1)$} \probpart{Find all points on $S$ at which
the tangent plane is parallel to the $x y$ plane.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{86}}{FINAL}{8}{}
\prob{Consider the curve $C: x = t, y =\frac{1}{t}, z = \ln t,$
and the line $L : x = 1+\tau , y = 1 + 2 \tau, z = -\tau.$ The
curve and the line intersect at the point $P = (1,1,0)$. Let
$\vec{v}$ be a unit vector tangent to $C$ at $P, \vec{w}$ a unit
vector tangent to $L$ at $P$. Compute the cosine of the angle
$\theta$ between $\vec{v}$ and $\vec{w}$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{87}}{\pr{1}}{4}{}
\prob{Consider the function $f(x,y,z) = 1 - 2 x^2 - 3 y^4.$}
\probpart{Find a unit vector that points in the direction of
maximum increase of $f$ at the point $R = (1,1,1).$}
\probpart{Find the outward unit normal to the surface $f = -4$ at
any point of your choice (clearly indicate your choice near your
answer).}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{87}}{FINAL}{6}{}
\prob{A particle moves with velocity $\vec{v}$ that depends on
position $(x,y). \vec{v} = (a+y) \vi + (-x + y) \vj.$ At $t = 0$
the particle is at $x = 1, y = 0$. Where is the particle at $t = 1
?$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{88}}{\pr{2}}{1}{}
\prob{Given $f = x y \sin z$ and $\vec{F} = (xy) \vi + (e^{y z}
\vj + (z^2) \vk$, evaluate:} \probpart{$\del f = grad(f) \mbox{
at } (x,y,z) = (1,2,3)$} \probpart{$\del \cdot \vec{F} \mbox{ at }
(x,y,z) = (1,2,3) $}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{88}}{\pr{3}}{1}{}
\prob{Curve $C$ is the line of intersection of the paraboloid$z =
x^2 + y^2 $ and the plane $z = x+\frac{3}{4}.$ The positive
direction on $C$ is the counterclockwise direction direction
viewed from above, i.e. from a point $(x,y,z)$ with $z > 0$.
Calculate the length of the curve $C$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SUMMER 1990}{\pr{1}}{1}{}
\prob{A parallelogram $ABCD$ has vertices at $A(2,-1,4),
B(1,0,-1), C(1,2,3)$ and $D$.} \probpart{Find the coordinates of
$D$.} \probpart{Find the cosine of the interior angle at $B$.}
\probpart{Find the vector projection of $\overrightarrow{BA}$
onto $\overrightarrow{BC}$} \probpart{Find the area of $ABCD.$}
\probpart{Find an equation for the plane in which $ABCD$ lies.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SUMMER 1990}{\pr{1}}{1}{}
\prob{Given the function $f(x,y) = e^{-x^2} + y - e^y:$}
\probpart{Compute the directional derivative at $(1,-1)$ in the
direction of the origin;} \probpart{Find all relative extreme
points and classify them as maximum, minimum, or saddle points;}
\probpart{Give the linearization of $f$ about $(1,-1)$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SUMMER 1990}{\pr{1}}{5}{3}
\prob{The position vector of a particle \[ \vec{R}(t) \mbox{ is
given by } \vec{R}(t) = t \cos t \vi + t \sin t \vj +
\brc{\frac{2 \sqrt{2}}{3}} t^{\frac{3}{2}}\vk
\]} \probpart{Find the velocity and acceleration of the particle at $t =
\pi$}\probpart{Find the total distance travelled by the particle
in space from $t = 0$ to $t = \pi$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SUMMER 1990}{\pr{1}}{4}{}
\prob{Find $\vec{T}, \vec{N}, \vec{B}$ and $\kappa$ at $t=0$ for
the space curve defined by \[ \vec{R}(t) = 2 \cos t \vi + 2 \sin
t \vj + t\vk
\]}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{90}}{FINAL}{10}{}
\prob{Find t e shortest distance from the plane $3x + y - z = 5$
to the point $(1,1,1)$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{90}}{\pr{1}}{1}{}
\probb{Find the equation of the plane $P$ which contains the
point $R = (2,1,-1)$ and is perpendicular to the straight line
$L: x = -1 + 2 t, y = 5 - 4t, z =t$.} \probpart{Find the point of
intersection of the lint $L$ and the plane $P$.} \probpart{Use b)
to find the distance of the point $R$ from the line $L$. }
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{UNKNOWN 1990}{UNKNOWN}{?}{}
\probb{Determine the rate of change of the function $f(x,y,z) =
e^x \cos y z$ in the direction of the vector $A = 2 \vi + \vj -
2\vk$ at the point $(0,1,0).$} \probpart{Determine the equation
of the plane tangent to the surface $e^x \cos y z = 1$ at the
point $(0,1,0).$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{90}}{\pr{1}}{2}{}
\probb{Find a \underline{unit} vector which lies in the plane of
$\vec{a}$ and $\vec{b}$ and is orthogonal to $\vec{c}$ if \[
\vec{a} = 2\vi - \vj + \vk, \vec{b} = \vi + 2 \vj - \vk , \vec{c}
= \vi + \vj - 2 \vk
\]} \probpart{Find the vector projection of $\vec{b}$ onto $\vec{a}.$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{90}}{\pr{1}}{3}{}
\prob{Show that the following are true} \probpart{$\brc{\vec{a}
\cdot \vi }^2 + \brc{\vec{a} \cdot \vj }^2 +\brc{\vec{a} \cdot \vk
}^2 = |\vec{a}|^2$} \probpart{$|\vec{a} \times \vec{b}|^2 +
(\vec{a} \cdot \vec{b})^2 = |\vec{a}|^2 |\vec{b}|^2$}
\probpartnn{(hint: use the angle between them)}
\probpart{$|\vec{a}|\vec{b} - |\vec{b}|\vec{a}$ is orthogonal to
$|\vec{a}|\vec{b} + |\vec{b}|\vec{a}$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{90}}{\pr{1}}{4}{}
\probb{Find $\vecv$ and $\vec{a}$ for the motion \[ \vec{R}(t) = t
\vi + t^3 \vj \]} \probpart{Sketch the curve including $\vecv,
\vec{a}.$} \probpart{Find the \underline{speed} at $t = 2.$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{90}}{\pr{1}}{5}{}
\prob{Let $\vec{R}(t) = (\cos 2 t) \vi + (\sin 2 t) \vj + t
\vk.$} \probpart{Find the length of the curve from $t = 0$ to $t =
1. $} \probpart{Find the unit tangent $\vec{T}$, the principal
unit normal $\vec{N}$ and the curvature $\kappa$ at $t =1$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{90}}{FINAL}{1}{}
\prob{Given the surface $z = x^2 + 2 y^2.$ At the point $(1,1)$
in the x-y plane:} \probpart{determine the direction of greatest
increase of z} \probpart{determine a unit normal to the surface.}
\probnn{Given the vector field $\vec{F} = 6 x y^ z \vi - 2 y^3 z
\vj + 4 z \vk, $} \probpart{calculate its divergence}
\probpart{use the divergence theorem to calculate the outward
flux of the vector field over the surface of a sphere of unit
radius centered at the origin.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{92}}{\pr{1}}{1}{}
\prob{Given the vectors \[ \vec{A} = \vi + \vj + \vk \] \[
\vec{B} = \vi + 2 \vj + 3 \vk
\] \[ \vec{C} = \vi - 2 \vj + \vk \] where $\vi, \vj \and \vk$ are mutually perpendicular unit
vectors.}\probnn{Evaluate} \probpart{$\vec{A} \cdot \vec{B}$}
\probpart{$\vec{A} \times \vec{B}$} \probpart{$\brc{\vec{A} \times
\vec{B}} \cdot \vec{C}$} \probpart{$\brc{\vec{A} \times \vec{B}}
\times \vec{C}$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{92}}{\pr{1}}{4}{}
\prob{Consider the plane $x + 2 y + 3 z = 17$ and the line
through the points $P: (0,3,4) \and Q:(0,6,2).$} \probpart{Is the
line parallel to the plane? Five clear reasons for your answer. }
\probpart{Find the point of intersection, if any, of the line and
the plane. }
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{92}}{\pr{1}}{2}{}
\prob{The acceleration of a point moving on a curve in space is
given by $\vec{a} = -\vi b \cos t - \vj c \sin t + 2 d \vk$ where
$\vi, \vj, \and \vk$ are mutually perpendicular unit vectors and
b,c and d are scalars. Also, the position vector $\vec{R}(t)$ and
velocity vector $\vec{v}(t)$ have the initial values \[
\vec{R}(0) = \vi(b+1), \vec{v}(0) = \vj c
\] Find $\vec{R}(t) \and \vecv(t)$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{92}}{\pr{1}}{5}{}
\prob{Consider the curve \[ \vec{R}(t) = 3 \vi + \vj \cos t + \vk
\sin t, 0 \leq t \leq 2 \pi
\] where $\vi, \vj \and \vk$ are mutually perpendicular unit
vectors.}\probpart{Sketch and describe the curve in words.}
\probpart{Determine the unit tangent, principal normal and
binormal vectors ($\vec{T}, \vec{N} \and \vec{B}$) to the curve
at the point $t = \frac{\pi}{2}$} \probpart{Sketch the vectors
$\vec{T}, \vec{N} \and \vec{B}$ at $t = \frac{\pi}{2}$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{92}}{\pr{1}}{6}{}
\prob{The position vector of a point moving along a curve is \[
\vec{R}(t) = t \vi + e^{2 t} \vj\] where $\vi \and \vj$ are
mutually perpendicular unit vectors and $t$ is time. The
acceleration vector $\vec{a}$ at the time $t = 0$ can be written
as \[ \vec{a}(0) = c \vec{T} + d \vec{N} \] where $\vec{T} \and
\vec{N}$ are the unit tangent and principal normal vectors to the
curve at the time $t = 0.$ } \probnn{Find the scalars $c \and d$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{92}}{FINAL}{1}{}
\prob{A point is moving on a spiral given by the equation \[
\vec{R}(t) = e^t \cos t \vi + e^t \sin t \vj
\] where $\vi \and \vj$ are the usual mutually perpendicular unit vectors.
Find}\probpart{The speed (the magnitude of the velocity) of the
point at $t = 0.$} \probpart{The curvature of the spiral at $t
=0$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SUMMER 1992}{PRELIM 6/30}{1}{}
\prob{Find the equation of the plane which passes through the
points \[ A(0,0,0), B(-1,1,0) \and C(-1,1,1). \]}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SUMMER 1992}{PRELIM 6/30}{5}{}
\prob{A point $P$, starting at the origin $(0,0,0)$ is moving
along a smooth curve. At any time, the distance $s$ travelled by
the point from the origin, is observed to be \[ s = 2 t\] Also,
the unit tangent vector to the curve, as this point, is \[
\vec{T} = -\frac{\sin t}{2}\vi + \frac{\cos t}{2} \vj +
\frac{\sqrt{3}}{2} \vk
\]} \probpart{Find the acceleration $\vec{a}$ of $P$ as a function of
time.}\probpart{Find the position vector $\vec{R}(t)$ of $P$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SUMMER 1992}{FINAL}{6}{}
\prob{A point $P$ is moving on a curve defined as}
\probpartnnn{$x(t) = \cos \alpha t$} \probpartnnn{$y(t) = 2 t$}
\probpartnnn{$z(t) = 3 \cos t + 6 t + 3(\alpha - 1) t^2$}
\probnn{Find value(s) of $\alpha$ such that the curve defined
above lies in a plane for all $0 \leq t \leq \infty$.}
\probnn{Hint: The idea  of torsion of a curve should be useful
here!}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{92}}{\pr{1}}{3}{}
\prob{Let $P_1 (-1,0,-1), P_2 (1,1,-1) \and P_3(1,-1,1)$ be three
points and let $\vec{A} = \overrightarrow{P_1 P_2} = 2 \vi + \vj
\and \vec{B} = \overrightarrow{P_1 P_3} = 2 \vi - \vj + 2 \vk.$}
\probpart{Find a vector perpendicular to the plane containing
$\vec{A} \and \vec{B}$.} \probpart{Find the area of the
parallelogram whose edges are $\vec{A} \and \vec{B}$.}
\probpart{Find the equation of the plane passing through the
points $P_1, P_2 \and P_3.$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{92}}{\pr{1}}{4}{}
\prob{Let $P_1 (-1,0,-1), P_2 (1,1,-1) \and P_3(1,-1,1)$ be three
points and let $\vec{A} = \overrightarrow{P_1 P_2} = 2 \vi + \vj
\and \vec{B} = \overrightarrow{P_1 P_3} = 2 \vi - \vj + 2 \vk.$}
\probpart{Find the distance from the point $(1,1,1)$ to the plane
passing through the points $P_1, P_2 \and P_3.$ } \probpart{Find
the equation of the line passing through the point $P_3$ and
parallel to the line passing through $P_1 \and P_2.$}
\probpart{Find the vector projection of $\vec{A}$ in the
direction of $\vec{B}$ and the scalar component of $\vec{A}$ in
the direction of $\vec{B}$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{92}}{\pr{1}}{5}{}
\prob{Let $\vec{u} \and \vec{v}$ be two given vectors. The vector
projection of $\vec{u}$ in the direction of $\vec{v}$ is
$\frac{(\vec{u} \cdot \vec{v})}{(\vec{v} \cdot \vec{v})} \vecv$.
Consider the vector $\vec{w} = \vec{u} - \frac{(\vec{u} \cdot
\vec{v})}{(\vec{v} \cdot \vec{v})} \vecv$. By taking the scalar
product of $\vec{w}$ with $\vec{v}$ show that $\vec{w}$ is
perpendicular (orthogonal) to $\vec{v}.$
$$\centerline{\epsfxsize=2.4in \epsfbox{6_1_346_01.jpg}}$$}
%------------------------------------------------------------------------------------------------------picture p.346
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{92}}{\pr{2}}{2}{}
\prob{Find all points $(x,y,z)$ which lie on the intersection of
the planes \[ x+y+z = 6, -x + 2 z = 1, y + 3 z = 7 \] Is this set
of points a single point, a line or a plane?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{92}}{\pr{2}}{3}{}
\prob{A point move on a space curve with the position vector \[
 \vecv(t) = e^t \cos t \vi + e^t \sin t \vj + 2 \vk\] Find the velocity $\vecv$, speed, unit tangent vector $\vec{T}$, unit principal normal $\vec{N}$, acceleration $\vec{a}$ and curvature $\kappa$ as functions of time. Also check that $\vec{N}$ is perpendicular to $\vec{T}$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{92}}{\pr{2}}{3?}{}
\prob{Determine the arc-length, $\int\limits_C d s$ of the curve
$C$ (a cycloid) given by: $r(t) = (t - \sin t) \vi + (1- \cos t)
\vj, 0 \leq t \leq 2 \pi$ (see figure below).
$$\centerline{\epsfxsize=2.4in \epsfbox{6_1_346_02.jpg}}$$}
%------------------------------------------------------------------------------------------------------picture p.346
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{93}}{\pr{1}}{4}{}
\prob{The four corners of a parallelopiped are given as $(1,1,1),
(1,4,2), (4,2,3) \and (1,1,4) $ in xyz-space. Using $(1,1,4)$ as
the common point of three vectors lying along the parallelopiped's
edges, calculate the volume of the parallelopiped.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{92}}{FINAL}{2}{}
\prob{A particle is moving along the positive branch of the curve
$y = 1+x^2$ and its x coordinates is controlled as a function of
time according to $x(t) = 2 t.$ Find} \probpart{The tangential
component of the particle's acceleration, $a_T,$ at time $t =0$.}
\probpart{The normal component of the particle's acceleration,
$A_N$, at time $t=0$.} \probpart{The radius of curvature $\rho$
of the curve, along which the particle is moving, at the point
$(0,1)$. Hint: $|\tilde{a}|^2 = a_N^2 + a_T^2, a_T = \frac{d
\tilde{v}}{d t}, a_N = \frac{|\tilde{v}|^2}{\rho}$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{93}}{\pr{1}}{6}{}
\prob{Find the equation of the plane that contains the
intersecting lines $L_1 \and L_2$ given by: \[ L_1: \left|
\begin{array}{l}
  x = 1+t \\
  y = 2+t \\
  z = 1+t
\end{array} \right. \; \; L_2:\left|
\begin{array}{l}
  x = 1-t \\
  y = 2-t \\
  z = 1
\end{array} \right.
\] Sketch the plane.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{93}}{\pr{1}}{5}{}
\prob{A line contains the two points $(1,2,3) \and (-2,1,4).$
Find parametric equations of the line and calculate the distance
from the line to the point $(5,5,5).$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{93}}{\pr{1}}{3}{}
\prob{Calculate the volume of the ellipsoid \[ x^2 +
\frac{y^2}{4} + \frac{z^2}{9} = 1
\] by imaging it to be comprised of a set of thin elliptical disks, of thickness $d z$, oriented parallel to the x-y plane.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{93}}{\pr{1}}{2}{}
\probb{Solve the initial value problem \[ \frac{d y}{d x} +
\frac{y}{x} = x^3
\] if $y =0$ when $x = 1$} \probpart{Consider a triangle $ABC$ with three vectors defined as \[ \vecv_1 = \overrightarrow{AB}, \vecv_2 = \overrightarrow{BC}, \vecv_3 = \overrightarrow{CA} \] From three points, one on each side of the triangle, draw vectors $\vec{w}_1, \vec{w}_2, \and \vec{w}_3$ in plane of the triangle. Each of these vectors is perpendicular to its side (i.e. $\vec{w}_1$ is perpendicular to $\overrightarrow{AB}$ and so on) with length equal to the length of the side and pointing out of the triangle. $$\centerline{\epsfxsize=2.4in \epsfbox{6_1_348.jpg}}$$}
%------------------------------------------------------------------------------------------------------picture p.348
\probparts{Find $\vec{w}_1, \vec{w}_2, \and \vec{w}_3$ in terms
of the components of $\vecv_1, \vecv_2, \and \vecv_3$.}
\probpart{Show that $\vec{w}_1 + \vec{w}_2 + \vec{w}_3 = 0$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{93}}{\pr{1}}{2}{}
\prob{$C$ is the curve given by \[ \vec{r}(t) = e^{-t} \cos t \vi
+ e^{-t} \sin t \vj + \sqrt{1-e^{-2 t}}\vk, \; (0 \leq t <
\infty).
\] Show that $C$ lies on the sphere $x^2 + y^2 + z^2 = 1$ and describe the curve with words and a
sketch.}\probnn{You may use the fact that $\cos t \vi + \sin t
\vj \; (0 \leq t \leq 2 \pi)$ is a parametrization of the unit
circle.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{93}}{\pr{1}}{3}{}
\prob{Find the equation of the plane that contains the
intersecting lines $L_1 \and L_2$ where: \[ L_1: \left|
\begin{array}{l}
  x = 1+t \\
  y = 1+t \\
  z = 1+t
\end{array} \right. \; \; L_2:\left|
\begin{array}{l}
  x = 1-t \\
  y = 1-t \\
  z = 1
\end{array} \right.
\]}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{93}}{\pr{1}}{4}{}
\prob{Find the equation of the plane through the points $(2,2,1)
\and (-1,1,-1)$ that is perpendicular to the plane $2 x - 3 y + z
= 3. $}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{93}}{\pr{1}}{5}{}
\prob{Consider a point $(x,y)$. Let $d_1$ be the distance from
$(x,y)$ to the line $x+y = 0$ and $d_2$ be the distance from
$(x,y)$ to the line $x-y = 0.$} \probnn{Given $d_1 d_2 = 1,$ find
the locus of all such points, i.e., say what the curve is and find
its equation. }
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{93}}{\pr{2}}{1}{}
\prob{A point $P$ is moving along a plane curve. The unit tangent
and principal normal vectors of this curve are, (for $t \geq 0$),
\[ \vec{T}(t) = -\vi \sin(t) + \vj \cos(t) \]  \[ \vec{N}(t) = -\vi \cos(t) - \vj \sin(t) \] (where $\vi \and \vj$ are the usual mutually perpendicular unit vectors), and the tangential component of the velocity vector of $P$, (the speed), is \[ \vec{v}_T = t.
\]} \probpart{Find the velocity vector $\vecv(t)$ of $P$.}
\probpart{Find the acceleration vector $\vec{a}(t)$ of $P$.}
\probpart{Find the tangential ($\vec{a}_T$) and normal
($\vec{a}_n$) components of the acceleration vector.}
\probpart{Find the radius of curvature $\rho(t)$ of the curve.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{94}}{\pr{1}}{1}{}
\prob{Find the distance from the point $(2,1,3)$ to the plane
which contains the points $(2,1,0), (0,1,1),(0,0,2)$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{94}}{\pr{1}}{2}{}
\prob{Find the point on the segment from $P_1 = (1,0,-1)$ to $P_2
= (4,3,2) $ which is twice as far from $P_2$ as it is from $P_1$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{94}}{\pr{1}}{3}{}
\prob{A particle moves on the sphere of radius $a$ centered at
the origin. Its position vector $\vec{r}(t)$ is a differentiable
function of the time, $t$. Show that the velocity vector
$\vecv(t)$ of the particle is always perpendicular to its position
vector, $\vec{r}(t).$ }
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{94}}{\pr{1}}{4}{}
\prob{A parallelogram, $P$, is determine by the two vectors $\vi
+ \vj + \vk \and 2 \vi - \vj - \vk.$} \probpart{What is the area
of $P$?} \probpart{What is the area of the orthogonal projection
of $P$ in the xy-plane?}\probpart{What is the area of the
orthogonal projection of $P$ in the xz-plane?} \probpart{What is
the area of the orthogonal projection of $P$ in the yz-plane?}
\probpart{What is the area of the orthogonal projection of $P$ in
the plane $x+y-z = 0$?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{94}}{\pr{2}}{1}{}
\prob{A point $P$ is moving along the spiral \[ x = e^t \cos (t)
\] \[ y = e^t \sin (t). \]} \probpart{Find the curvature of the given spiral at $t =
0.$}\probpart{The acceleration of $P$ is written as \[ \vec{a} =
a_T \vec{T} + a_N \vec{N}. \] Find $a_T$ and $a_N$ at $t = 0.$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{94}}{\pr{1}}{1}{}
\prob{Let \[ \vec{A} = 2 \vi - \vj + \vk \] \[ \vec{B} = \vi +
\vj + \vk
\] \[ \vec{C} = \vi + 2 \vj + \vk\]} \probpart{Find the vector projection of $A$ onto the direction of
$\vec{B}$.}\probpart{Show that $\vec{A} - proj_{\vec{B}}\vec{A}$
is perpendicular to $\vec{B}$.} \probpart{Find the area of the
parallelogram with edges $\vec{A} \and \vec{B}$.} \probpart{Find
the volume of the box with edges $\vec{A}, \vec{B} \and
\vec{C}.$} \probpart{Find the parametric equation of the line
through $(0,0,0)$ and parallel to the intersection of the planes
with normals $\vec{A} \and \vec{B}$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{94}}{\pr{1}}{2}{}
\prob{Let $\vec{a} \and \vec{b}$ be vectors. Show that}
\probpart{$|\va \times \vb|^2 + (\va \cdot \vb)^2 =
|\va|^2|\vb|^2,$ and} \probpart{that $|\va|\vb - |\vb|\va$ is
perpendicular to $|\va|\vb - |\vb|\va$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{94}}{\pr{1}}{3}{}
\prob{Graph $z = x^2 + y^2 + 1$ and label any intersection the
surface may have with any axis. Describe the curves that are the
intersections of the surface with the planes $z = $ constant $(z
> 1). $}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{94}}{\pr{2}}{1}{}
\prob{Consider the path traversed by a particle given
parametrically by $\vr(t) = (e^t \cos t) \vi + (e^t \sin t) \vj +
e^t \vk. $ Find the} \probpart{velocity vector} \probpart{speed}
\probpart{acceleration vector} \probpart{length of the path from
$t = 0$ to $t = \ln 4$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{94}}{FINAL}{1}{}
\prob{The level curves of the function $f(x,y,z) = z+x^2+y^2+1$
are:} \probpart{Hyperboloids} \probpart{Planes} \probpart{Cones}
\probpart{Paraboloids} \probpart{Spheres}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{94}}{FINAL}{3}{}
\prob{The vector projection of $(1,0,1,0)$ in the direction of
$(1,1,1,1)$ is:} \probpart{$\brc{-\frac{1}{2}, \frac{1}{2},
-\frac{1}{2}, \frac{1}{2}},$} \probpart{$(0,1,0,1),$}
\probpart{$\brc{\frac{1}{2}, \frac{1}{2}, \frac{1}{2},
\frac{1}{2}},$} \probpart{$(0,-1,0,-1)$}
\probpart{$\brc{\frac{1}{2}, -\frac{1}{2}, \frac{1}{2},
-\frac{1}{2}}$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{94}}{FINAL}{5}{}
\prob{Any non-zero vector perpendicular to the vectors $\vi + \vj
+ \vk \and \vi + 2 \vk $ is} \probpart{Perpendicular to $2 \vi +
\vj + \vk,$} \probpart{Parallel to $\vi+\vj+\vk,$}
\probpart{Perpendicular to $2\vi - \vj - \vk,$}
\probpart{Parallel to $2\vi + \vj + \vk,$} \probpart{Parallel to
$2 \vi - \vj - \vk$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{95}}{\pr{1}}{2}{}
\prob{Consider the planar curve \[ y^2 = 4x. \] Find parametric
equations of the following lines.} \probpart{Tangent to the above
curve at $P(1,2).$} \probpart{Normal to the above curve at
$O(0,0)$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{95}}{\pr{1}}{3}{}
\probb{Show that the points $\vtthree{1}{1}{0},
\vtthree{2}{1}{1}, \vtthree{3}{3}{-1} \and \vtthree{4}{3}{0}$ are
the vertices of a parallelogram. } \probpart{What is the area of
this parallelogram?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{95}}{\pr{1}}{5}{}
\prob{The surface $S$ drawn below can be described in two ways,
i.e.} \probpartnn{as $z = f(x,y) = 1-x^2 - y^2, -1 \leq x \leq 1,
-1 \leq y \leq 1 $} \probpartnn{or $g(x,y,z) = z + x^2 + y^2 =1,
-1 \leq x \leq 1, -1 \leq y \leq 1 $} \probnn{Evaluate and sketch
the gradient fields $\del f \and \del g.$ Explain the
relationship between these two vector
fields.$$\centerline{\epsfxsize=2.4in \epsfbox{6_1_351.jpg}}$$}
%------------------------------------------------------------------------------------------------------picture p.351
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{95}}{\pr{2}}{2}{}
\prob{Let \[ \vtthree{x(t)}{y(t)}{z(t)} \] be a space curve; let
$\vv(t)$ be the velocity vector and $\va (t)$ the acceleration
vector.} \probpart{Give the formula which gives the curvature of
the curve in terms of $\vv \and \va.$} \probpart{By differenting
$|\vv|^2 = \vv \cdot \vv,$ find a formula for $\frac{d \vv}{d t}$
in terms of $\vv \and \va$} \probpart{If at some instant we have
\[ |\vv| = 3 \; m/s, \frac{d |\vv|}{d t} = 4 \; m/s^2, |\va| = 5 \; m/s^2\] what is the radius of curvature in meters.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{95}}{\pr{1}}{1}{*}
\prob{This is a two-dimentional problem. Consider the parabola \[
y^2 = 4 x \mbox{ and the point } P(1,2) \mbox{ on it. } \]}
\probpart{Find an unit vector $\vt$ that is tangential to the
parabola at $P$.} \probpart{Find the equation of the tangent line
to the parabola at $P$. Any correct form of the equation is
acceptable.} \probpart{Find an unit vector $\vn$ that is normal
to the parabola at $P$.} \probpart{Find the equation of the
normal line to the parabola at $P$. Any correct form of the
equation is acceptable.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{95}}{\pr{1}}{1}{*}
\probb{For $f = x^2 + 8 y^2,$ show that $(4,2)$ lies on the level
curve $f(x,y) = 48.$ Sketch this level curve.} \probpart{Find the
vector field $\del f$} \probpart{Evaluate $\del f$ at $(x,y) =
(\sqrt{48},0), (4,2), (4,-2), (0, \sqrt{6})$ and sketch these
vectors, showing very clearly their relation to the level curve.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{95}}{\pr{1}}{2}{}
\prob{Consider two straight lines in space given by the
equations: \[ L_1 : \left| \begin{array}{l}
  x = 2+t \\
  y = 2+t \\
  z = -t
\end{array} \right. \; -\infty \leq t \leq \infty\] \[ L_2 : \left| \begin{array}{l}
  x = 3+u \\
  y = -2 u \\
  z = 1+u
\end{array} \right. \; -\infty \leq u \leq \infty\]} \probpart{Do these lines intersect? If so, find the coordinates of the point of
intersection.}\probpart{Find a vector $\vu$ along $L_1$ and a
vector $\vv$ along $L_2.$} \probpart{Find, if possible, the
equation of the plane that contains the lines $L_1 \and L_2$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{95}}{\pr{1}}{4a}{*}
\prob{Describe the set of points defined by the equations \[
\begin{array}{rcl}
  x^2 + y^2 + z^2 & \leq & 4 \\
  z & \leq & 1
\end{array}
\] Also, draw a sketch showing this set of points.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{95}}{FINAL}{5}{*}
\prob{A point $P$ is moving on a plane curve with the position
vector \[ \vr(t) = x(t) \vi + y(t) \vj , t \geq 0 \] where $t$ is
time and $\vi \and \vj$ are the usual orthogonal Cartesian unit
vectors. The position components $x(t) \and y(t)$ satisfy the
equations \[ t \frac{d x}{d t} + x = t^2, x(0) = 0 \] \[ \and
\frac{d^2 y}{d t^2} - 6\frac{d y}{d t} + 9 y = 0, y(0) = 0,
\frac{d y}{d t} (0) = 1
\]} \probpart{Find $x(t)$ as an explicit function of time.}
\probpart{Find $y(t)$ as an explicit function of time.}
\probpart{Find $\vv(t) = \frac{d \vr}{d t},$ the velocity of $P$
as a function of time.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{96}}{FINAL}{17}{*}
\prob{A bug flies around the room so that at time $t$, the
position of the bug is given by $x = t^2, y = t^{\frac{3}{2}}, z
= t^2.$ The velocity at time $t = 1$ is} \probpart{10.25}
\probpart{$2\vi + \frac{3}{2} \vj + 2 \vk$}
\probpart{$\vi+\vj+\vk$} \probpart{3} \probpart{none of the above}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{96}}{FINAL}{18}{*}
\prob{The speed of the bug above at time $t = 1$ is}
\probpart{40} \probpart{19} \probpart{3} \probpart{$\vi + 3 \vj +
8\vk $} \probpart{none of the above}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{96}}{FINAL}{19}{*}
\prob{The position of the bug above at time $t = 1$
is}\probpart{$\sqrt{3}$} \probpart{40} \probpart{19}\probpart{3}
\probpart{none of the above}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{96}}{FINAL}{25}{*}
\prob{A cannon fires a cannonball at an angle of 45 degrees from
horizontal. The cannonball lands 1000 meters away. Taking
Newton's gravitational constant $g$ to be 10 meters per second
squared, the speed of the cannonball when leaving the cannon in
meters per second is}\choice{10}{$10
\sqrt{10}$}{100}{$\frac{2000}{\sqrt{2}}$}{none of the above}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{96}}{FINAL}{20}{*}
\prob{The projection of the vector (1,0,1,0) in the direction of
(1,1,1,1) is} \choice{$\brc{-\frac{1}{2},
\frac{1}{2},-\frac{1}{2},\frac{1}{2}}$}{(0,1,0,1)}{$\brc{\frac{1}{2},
\frac{1}{2},\frac{1}{2},\frac{1}{2}}$}{(0,-1,0,1)}{none of the
above}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{96}}{FINAL}{24}{*}
\prob{Let $P = (1,1,1), Q =(1,0,0), R = (0,1,0).$ Then the
equation of the plane in $\Re^3$ containing the triangle $PQR$
is} \choice{$x+y-z = -1$}{$x - y + z = 1$}{$-x + y - z = -1$}{$x
+y-z = 1 $}{none of the above}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{96}}{FINAL}{30}{*}
\prob{If $\vu, \vv, \vw$ are vectors in $\Re^3$, then $\vu \cdot
(\vv \times \vw) = (\vw \times \vv) \cdot \vu $ (T/F)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{96}}{FINAL}{1 MAKE-UP}{}
\prob{In this problem $f(x,y) = x - y^2.$} \probpart{Sketch the
level curve $f(x,y) = -7$} \probpart{Evaluate $\del f(2,3)$ and
sketch it on the graph, showing the relation to the level curve.}
\probpart{Find the to the right flux of $\del f$ across the
segment $0 \leq y \leq 5, x = 0$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{97}}{\pr{3}}{4}{}
\prob{Evaluate the line integral \[ \int_C \frac{z y d x + z x d
y + (z - x y) d z }{z^2}
\] where $C$ is the curve given by parametric equations $x(t) = \cos(\pi t), y(t) = \sin (\pi t), z(t) = t, (1 \leq t \leq 2).$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SUMMER 1992}{PRELIM 6/30}{3}{}
\prob{A point is moving along a curve given by the parametric
equations \[ x(t) = t \] \[ y(t) = 2 t^2 \] Find, as functions of
time $t$ } \probpart{The velocity of the point, $\vv$}
\probpart{The acceleration of the point, $\va$} \probpart{The
curvature $\kappa$ of the curve} \probpart{If $\va = a_N \vec{N}
+ a_T \vec{T}, $ where $\vec{N}$ is the principal unit normal and
$\vec{T}$ is the unit tangent vector to the curve at some point
on it, find $a_N \and a_T.$ }
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{95}}{\pr{2}}{1}{}
\prob{Consider the spiral parametrized by \[ t \mapsto
\vttwo{e^{-t} \cos t}{e^{-t} \sin t} \; \; 0 \leq t < \infty.
\]} \probpart{Sketch the curve.} \probpart{Find its length or show that it has infinite length.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{94}}{\pr{2}}{1}{}
\prob{Find the arc length parametrization of the space curve: \[
\vr(t) = \cos(2 t) \vi + \sin (2 t) \vj + \frac{2}{3}
t^{\frac{3}{2}} \vk, \mbox{ with } 0 \leq t \leq 5.
\]}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SUMMER 1995}{\pr(1)}{1}{}
\prob{The surface \[ z = f(x,y) = y^2 - x^2; -2 \leq x \leq 2, -2
\leq y \leq 2
\] is shown below along with its normal vectors.} \probpart{Sketch the contour lines of the surface in the $(x,y)$ plane, i.e. draw the curves such that $z =$ constant, for example $z =
-2,-1,0,+1,+2.$} \probpart{On your sketch for part (a) sketch the
vector field $\del f.$ } \probpart{Find an expression for the
unit normal vectors $\vec{n}$ of the
surface.$$\centerline{\epsfxsize=2.4in
\epsfbox{6_1_355_01.jpg}}$$}
%------------------------------------------------------------------------------------------------------picture p.355
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{92}}{FINAL}{2}{}
\prob{Consider the curve $C: \vr(t) = t \cos t \vi + t \sin t \vj
+ t \vk, 0 \leq t \leq 4 \pi, $ which corresponds to the conical
spiral shown below.} \probpart{Set up, but so not evaluate, the
integral yielding the arc-length of $C$.} \probpart{Compute
$\int_C (y+z) d x + (z + x) d y + (x + y) d z.$
$$\centerline{\epsfxsize=2.4in \epsfbox{6_1_355_02.jpg}}$$}
%------------------------------------------------------------------------------------------------------picture p.355
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{94}}{FINAL}{2}{}
\prob{A bug flies around the room along a path parametrized by $x
= t^2, y = t^{\frac{3}{2}}, z = t^2 .$ If the temperature at any
point (x,y,z) is given by $T(x,y,z) = x^2 y + z^2,$ the rate at
which the bug feels the temperature change when $t = 1$ is}
\choice{3}{-3}{$\frac{19}{2}$}{0}{$\frac{15}{2}$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{94}}{\pr{1}}{1}{*}
\prob{$C$ is the line segment from (0,1,2) to (2,0,1).
$$\centerline{\epsfxsize=2.4in \epsfbox{6_1_356.jpg}}$$}
%------------------------------------------------------------------------------------------------------picture p.356
\probpart{which of the following is a parametrization of $C$?}
\probparts{$x = 2 t , y = 1-t, z = 2- t, 0 \leq t \leq 1$}
\probparts{$x = 2 - 2t, y = -2 t, z = 1 - 2t, 0 \leq t \leq
\frac{1}{2}$} \probparts{$x = 2 \cos t, y = \sin t, z = 1 + \sin
t, 0 \leq t \leq \frac{\pi}{2} $} \probpart{evaluate $\int_C 3 z
\vj \cdot d \vr $}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{96}}{\pr{1}}{1}{*}
\probb{Evaluate $\int^{(4,0,2)}_{(0,0,0)} 2 x z^3 d x + 3 x^2 z^2
d z $ on any path.} \probpart{Write parametric equations for the
line segment from (1,0,3) to (2,5,0).}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{92}}{\pr{1}}{3}{}
\prob{Given a plane $x - 5 y + z = 21$ and a point $R$ with
coordinates (1,2,3), find} \probpart{The parametric equations of
a line perpendicular to the plane and passing through $R$.}
\probpart{The point of intersection of the line and the plane.}
\probpart{The distance from $R$ to the plane.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{97}}{\pr{3}}{3}{}
\prob{Consider the sphere $x^2 + y^2 + z^2 = 25.$}
\probpart{Express the equation of the sphere in cylindrical
coordinates $(r,\theta,z)$ and find volume inside it by
evaluating a triple integral in cylindrical coordinates.}
\probpart{Now consider the region that you get by starting with
the solid interior of the sphere as before, and removing the
points which are contained inside the cone $z = \sqrt{x^2 +
y^2}.$ This means that our new region consists of points having
$x^2 + y^2 + z^2 \leq 25, z \leq \sqrt{x^2 + y^2}.$ Find the
volume of this region by evaluating a triple integral spherical
coordinates $(\rho,\phi,\theta).$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SUMMER 1992}{PRELIM 6/30}{6}{}
\prob{A consider is described by the parametric equations \[ x =
2 \cos t
\] \[ y = 2 \sin t \] A point $P$ inside the circle has coordinates (1,1). The line, normal to the circle, through $P$, intersects the circle at two points $Q_1 \and Q_2. Q_1$ is the point nearer to
$P$.}\probpart{Find the vector $\vec{N}$ along the line $PQ_1$}
\probpart{Find the parametric equations of the line segment
$PQ_1.$} \probpart{Find the distance $PQ_1.$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SUMMER 1990}{\pr{1}}{2}{}
\probb{Find the distance between the point $P =(0,0,0)$ and the
line $L$ defined parametrically by \[ X = t+1\] \[ Y = t+1 \] \[
Z = t\]} \probpart{Find an equation of the line through $P$ that
is perpendicular to the line $L$.}
\end{problem}
