Each of these problems is intended to teach you something,
to heighten your understanding of the subject.
Be sure to not simply “do” the problem as if it were a task to complete,
but to reflect on its purpose and learn something from it.
Describe the set of all points
that are equidistant (the same distance)
from the points \((1,2,3)\) and \((-6,5,4).\)
Can you find an equation that corresponds to this set?
I.e. can you find the equation that relates
any \(x\) and \(y\) and \(z\)
such that \((x,y,z)\) is equidistant from those points?
Describe the set of all points
that are twice the distance to \((1,2,3)\)
as they are to \((-6,5,4).\)
Can you find an equation that corresponds to this set?
What is an equation for the plane
that crosses the \(x\)-axis at \(3,\)
crosses the \(y\)-axis at \(5,\)
and crosses the \(z\)-axis at \(11?\)
What is an equation for the sphere
that is centered at \((1,2,3)\)
and passes through the point \((-6,5,4)?\)
What is the shortest distance from the point \((1,2,3)\)
to the plane \(2x-3y+7z = 11?\)
Given two parallel planes
\( ax+by+cz = p\) and \( ax+by+cz = q,\)
find a formula for the shortest distance between them.
The earth is approximately a sphere, but only approximately.
Due to the earth’s rotation and
centrifugal force,
it’s “bulging” at the equator and would be more accurately described
as an ellipsoid (or oblate spheroid).
The current World Geodesic System standard,
WGS-84,
establishes the distance from the earth’s center to the equator as 6,378.1370 km,
whereas the distance from the earth’s center to either pole is 6,356.7523 km.
Write an equation that describes a surface in three-dimensional space
that models the ellipsoidal surface of this earth
where its center is at the origin \((0,0,0)\)
and the north pole lies on the \(z\)-axis.
Are there any triples \((a,b,c)\) that represent the same point in space
whether interpreted as a location in rectilinear or cylindrical coordinates?
What about in rectilinear and spherical coordinates?
The convention when defining spherical coordinates
does not match up with our convention for defining
GPS coordinates (longitude and latitude) on earth.
Like how we have formulas to convert
between rectangular coordinates and spherical coordinate,
we should develop formulas to convert
between rectangular coordinates and GPS coordinate.
Look up how GPS coordinates describe a point on the earth.
Suppose the earth is oriented in space such that
the center of the earth is at the origin,
the positive \(z\)-axis is aligned with the north pole,
the positive \(x\)-axis passes through the point
with GPS coordinates \((0,0)\) on the earths surface,
and the \(y\)-axis is oriented in accordance with the right-hand rule.
What must be true of a triple \((x,y,z)\) in rectangular coordinates
to guarantee that point is on the surface of the earth?
(instead of within the earth, or floating out in space, for instance)
Given a triple \((x,y,z)\) in rectangular coordinates
that corresponds to a point on earth,
what is the latitude \(\varphi\) and longitude \(\lambda\)
of that point?
Given the GPS coordinates \(\varphi\) and \(\lambda\)
of a point in space on the earth’s surface,
what are it’s rectangular (absolute) coordinates?
Where on earth’s surface does the positive \(y\)-axis cross?
Any two diagonals of a cube intersect in the center of the cube.
What is the acute angle at which two diagonals of a cube intersect?
These are some equations that are usually referred to as
“properties of dot products” or “properties of cross products.”
Verifying these properties algebraically would be tedious; don’t do that.
Instead, come up with geometric intuition on why each of these properties holds true.
A plane is capable of flying at a speed of 180 km/h in still air.
The pilot takes off from an airfield and heads due north
according to the plane’s compass.
After 30 minutes of flight time, the pilot notices that,
due to the wind, the plane has actually traveled 80 km
at an angle 5° east of north.
What is the wind velocity?
In what direction should the pilot have headed to reach the intended destination?
Suppose the three coordinate planes are all mirrored
and a light ray given by the vector \(\mathbf{a}= \langle a_1, a_2, a_3 \rangle\)
first strikes the \(xz\)-plane, as shown in the figure.
Use the fact that the angle of incidence equals the angle of reflection
to show that the direction of the reflected ray
is given by \(\mathbf{b} = \langle a_1, -a_2, a_3\rangle.\)
Deduce that, after being reflected by all three mutually perpendicular mirrors,
the resulting ray is parallel to the initial ray.
A projectile is fired from the origin
with angle of elevation \(\alpha\) and initial speed \(v_0.\)
Assuming that air resistance is negligible
and that the only force acting on the projectile is gravity \(g,\)
the position vector of the projectile is
\[\mathbf{r}(t) = \bigl(v_0\cos(\alpha)t\bigr)\mathbf{i} + \bigl(v_0\sin(\alpha)t - \frac{1}{2}gt^2\bigr) \mathbf{j}.\]
The maximum horizontal distance the projectile can travel is \(R = v_0^2/g\)
and is achieved when \(\alpha = 45°.\)
At what angle should the projectile be fired to achieve maximum height
and what is the maximum height?
Fix the initial speed \(v_0\)
and consider the parabola \(x^2 + 2Ry - R^2 = 0\)
whose graph is shown in the figure.
Show that the projectile can hit any target inside or on the boundary
of the region bounded by the parabola and the \(x\)-axis,
and that it can’t hit any target outside this region.
Suppose that the gun is elevated to an angle of inclination \(\alpha\)
in order to aim at a target that is suspended at a height \(h\)
directly over a point \(D\) units downrange.
The target is released at the instant the gun is fired.
Show that the projectile always hits the target, regardless of the value \(v_0,\)
provided the projectile does not hit the ground “before” \(D.\)
A projectile is fired from the origin down an inclined plane
that makes an angle \(\theta\) with the horizontal.
The angle of elevation of the gun and the initial speed of the projectile
are \(\alpha\) and \(v_0\) respectively.
Find the position vector of the projectile
and the parametric equations of the path of the projectile
as functions of the time \(t.\) (Ignore air resistance.)
Show that the angle of elevation \(\alpha\)
that will maximize the downhill range
is the angle halfway between the plane and the vertical.
Suppose the projectile is fired up an inclined plane
whose angle of inclination is \(\theta\).
Show that, in order to maximize the (uphill) range,
the projectile should be fired in the direction halfway
between the plane and the vertical.
In a paper presented in 1686,
Edmond Halley summarized the laws of gravity and projectile motion
and applied them to gunnery.
One problem he posed involved firing a projectile to hit a target
a distance \(R\) up an inclined plane.
Show that the angle at which the projectile should be fired to hit the target
but use the least amount of energy is the same as the angle in the previous part.
(Use the fact that the energy needed to fire the projectile is proportional to
the square of the initial speed, so minimizing the energy
is equivalent to minimizing the initial speed.)
A projectile of mass \(m\) is fired from the origin
at an angle of elevation \(\alpha\).
In addition to gravity, assume that air resistance provides a force
that is proportional to the velocity and that opposes the motion.
Then, by Newton’s Second Law,
the total force acting on the projectile satisfies the equation
\[
m\frac{\mathrm{d}^2\mathbf{R}}{\mathrm{d}t^2}
= -mg\mathbf{j} - k\frac{\mathrm{d}\mathbf{R}}{\mathrm{d}t}
\]
where \(\mathbf{R}\) is the position vector
and \(k \gt 0\) is the constant of proportionality.
Show that this equation can be integrated to obtain the equation
\[
\frac{\mathrm{d}\mathbf{R}}{\mathrm{d}t}
+ \frac{k}{m}\mathbf{R} = \mathbf{v}_0-gt\mathbf{j}
\]
where \(\mathbf{v}_0 = \mathbf{v}(0) = \frac{\mathrm{d}\mathbf{R}}{\mathrm{d}t}(0).\)
Multiply both sides of the equation in the last part
by \(\mathrm{e}^{(k/m)t}\) and show that
the left-hand side of the resulting equation
is the derivative of the product
\(\mathrm{e}^{(k/m)t}\mathbf{R}(s).\)
Then integrate to find an expression
for the position vector \(\mathbf{R}(r).\)
Find the curvature of the curve defined parametrically by these equations:
\[
x = \int\limits_0^t \sin\biggl(\frac{\pi}{2}\theta^2\biggr)\,\mathrm{d}\theta
\qquad
y = \int\limits_0^t \cos\biggl(\frac{\pi}{2}\theta^2\biggr)\,\mathrm{d}\theta
\]
In designing transfer curves
to connect sections of straight railroad tracks,
it’s important to realize that the acceleration of the train
should be continuous so that the reactive force
exerted by the train on the track is also continuous.
This will be the case if the curvature varies continuously.
A logical candidate for a transfer curve to join existing tracks
given by lines \(y=1\) for \(x\leq0\) and
\(y = \sqrt{2} - x\) for \(x \gt 1/\sqrt{2}\)
might be the function \(f(x) = \sqrt{1-x^2}\) for \(0 \lt x \lt 1/\sqrt{2},\)
whose graph is the arc of the circle shown in the figure.
It looks reasonable at first glance.
Show that the function
\[
F(x) = \begin{cases}
1 \quad&\text{ if } x \leq 0
\\ f(x) \quad&\text{ if } 0 \lt x \lt 1/\sqrt{2}
\\ \sqrt{2}-x \quad&\text{ if } x \geq 1/\sqrt{2}
\end{cases}
\]
is continuous and has continuous slope,
but does not have continuous curvature.
Therefore, \(f\) is not an appropriate transfer curve.
Find a fifth-degree polynomial to serve as a transfer curve
between the following straight line segments:
\(y=0\) for \(x\leq0\) and \(y=x\) for \(x \geq 1.\)
Could this be done with a fourth-degree polynomial?
Use a graphing calculator or computer
to sketch the graph of the “connected” function
and check to see that it looks like the one in the figure.