Euclidean Space and Vectors

startsection section10mm-.5 Discussion 1 Functions of Several Variables

 

In one variable calculus you considered functions of one independent variable, which were written symbolically in the form

y = f(x).

Here we will be focusing for the most part on functions of two independent variables, which can be written symbolically in the form

z= f(x,y).

Many of you have worked with functions of this form already, either at the end of a one-variable calculus course, or in physics, chemistry or biology. While the sciences may not use the standard mathematical symbolism, they provide a wealth of examples of functions. In this abbreviated discussion, we will explore what you know about functions.

One member of your group should be the scribe for the day. The scribes job is to write a summary of your responses to each of the following questions that should be handed in at the end of class.

startsection subsection10mm.5 Exercises

1.

You have all explored function of one variable in precalculus and calculus. What are some of the questions you asked about these functions? Discuss the different ways functions were represented as well as the mathematical techniques used to analyze these functions.

2.

What, if any, contact have you had with functions of more than one variable in a science? Provide examples of the functions where possible. It is sufficient to describe the relevant quantities in prose rather than symbols.

3.

How might the questions you asked about functions of one variable in calculus be relevant to functions of more than one variable? Are there other questions you might ask about functions of more than one variable?

4.

In order to help establish working groups, we need to know more about our learning styles and our mathematical strengths and weaknesses. Different students learn mathematics in different ways. Some students are more visual -- they can understand more from pictures and graphical displays of information. Others are more computational -- they learn better through applications and problem solving. Some students learn through patterns, that is, they see several examples and work through several more on their own until they can recognize general patterns. Others are able to understand general structures and rules and then are able to use computational techniques to fill in the details.

What are the mathematical strengths of each member of the group? What about the other strengths of group members, for example, who takes notes well; which members are willing to suggest ideas; who will ask questions when they don't understand? What strengths do you hope members of your group would contribute to the discussion?

startsection section10mm-.5 Discussion 2 The Coordinate System in ${\bf R}^{3}$

 

In this discussion, we will use the Euclidean coordinate system in ${\bf R}^{3}$ to describe subsets of ${\bf R}^{3}$. The sets will be presented either in symbols or in prose, and you will have to translate the description from one form into the other form and sketch the set. Several of the prose descriptions are presented in the form of a model from outside of mathematics.

startsection subsection10mm.5 Exercises

1.

For each of the following prose descriptions of subsets of ${\bf R}^{3}$, sketch the subset and give a symbolic description of the subset in set notation. For the last two parts of the problem, the context for the description of the set is provided after the description of the set. It will be helpful to understand the context before you answer the question.

(a)

The set of all points ``which are less than one unit from the origin.''

(b)

The set of all points ``inside a cube with sides of length two centered at the point (1,1,1).''

(c)

The set of all points ``which are less than two units from the point (1,2,-1).''

2.

For each of the following symbolic descriptions of subsets of ${\bf R}^{3}$, sketch the subset and give a prose description of the subset.

(a)

$\{(x, y, z)\; : \; 0 \leq x \leq 1,\ 0 \leq y \leq 2 \mbox{ and } 0 \leq z \leq 1-x \}.$

(b)

$\{(x, y, z)\; : \; 0 \leq x \leq 1,\ 0 \leq y \leq 1-x \mbox{ and } 0 \leq z \leq 1 + x + y \}.$

(c)

$\{(x, y, z)\; : \; x^2 + y^2 \leq 4 \}.$

  

Figure: The robot for exercise 3.

3.

In programming the motion of a computer controlled robotic arm, it is useful to have a mathematical description of the region of space that can be reached by the grasping end of the arm given the location of the base of the arm and the range of motion at each pivot. Here is a particular example to think about. (See Figure .)

The base of the arm is located on a pedestal of height h and both the lower portion and the upper portion of the arm have length 2h. The arm can rotate in a complete circle on its pedestal and the only constraint on the motion of the arm is that the angle between the lower portion of the arm and the upper portion of the arm must be less than or equal to $\pi/2$. In your calculations, assume that the pedestal is centered on the vertical axis and the base of the arm is located at (0,0,h).

(a)

What is the radius of the largest circle in the (x,y)-plane that can be reached by the grasping end of the arm?

(b)

The arm is designed to remove items from a conveyer belt on the (x,y)-planethat is located a distance of h units from the vertical axis. What is the length of the segment of the conveyer belt that is accessible to the grasping end of the arm?

startsection section10mm-.5 Discussion 3 Using the Language of Vectors

 

In this discussion, we will use the language of vectors to describe subsets of the plane and of space. In order to do this, we will use the correspondence we have established between vectors in the plane or in space and points in ${\bf R}^2$ or ${\bf R}^{3}$. Each vector ${\bf v}$ is placed with initial point at the origin, the point which corresponds to this vector is the endpoint of ${\bf v}$. Thus the vector ${\bf v}= (v_1,v_2)$ corresponds to the point $P = (v_1,v_2) \in {\bf R}^2$ and the vector ${\bf v} = (v_1,v_2,v_3)$ corresponds to the point $P = (v_1,v_2,v_3) \in {\bf R}^3$.

startsection subsection10mm.5 Exercises

In each of these exercises, use the correspondence above to describe in words the subset of ${\bf R}^2$ or ${\bf R}^{3}$ which corresponds to each set of vectors.

1.

Describe in words and sketch the set of points in the plane or in space which corresponds to each set of vectors. In each case, ${\bf u}$ and ${\bf v}$ are fixed vectors.

(a)

i.

$\{ c{\bf u}: \ c \in {\bf R}\}$

ii.

$\{ c{\bf u}: \ c \in [0,1] \}$

iii.

$\{ c{\bf u}: \ c > 0 \}$

iv.

$\{ c{\bf u}: \ c < 0 \}$

(b)

i.

$\{ {\bf v}+ c{\bf u}: \ c \in {\bf R}\}$

ii.

$\{ {\bf v}+ c{\bf u}: \ c \in [0,1] \}$

iii.

$\{ {\bf v}+ c{\bf u}: \ c > 0 \}$

iv.

$\{ {\bf v}+ c{\bf u}: \ c < 0 \}$

2.

Describe in words and sketch the set of points in ${\bf R}^{3}$ that corresponds to each set of vectors. In each case, ${\bf u}$, ${\bf v}$, and ${\bf w}$ are fixed vectors.

(a)

i.

$\{ c{\bf u}+ d{\bf v}: \ c,\ d \in {\bf R}\}$

ii.

$\{ c{\bf u}+ d{\bf v}: \ c, \ d \in [0,1] \}$

iii.

$\{ c{\bf u}+ d{\bf v}: \ c, \ d > 0 \}$

(b)

i.

$\{ {\bf w}+ c{\bf u}+ d{\bf v}: \ c,\ d \in {\bf R}\}$

ii.

$\{ {\bf w}+ c{\bf u}+ d{\bf v}: \ c, \ d \in [0,1] \}$

iii.

$\{ {\bf w}+ c{\bf u}+ d{\bf v}: \ c, \ d > 0 \}$

startsection section10mm-.5 Discussion 4 The Geometry of Lines and Planes

 

In this discussion, we want to use our descriptions of lines and planes to explore the possible geometric relationships between a line and a plane or between two lines in ${\bf R}^{3}$.

startsection subsection10mm.5 Exercises

1.

(a)

Suppose ${\cal L}$ is a line in ${\bf R}^{3}$ and ${\cal P}$ is a plane in ${\bf R}^{3}$, what are the possible geometric relationships between ${\cal L}$ and ${\cal P}$?

(b)

Explain how you would determine what the geometric relationship is between a given line and plane.

(c)

Determine the relationship between each of the following lines and planes

i.

Suppose ${\cal L}$ is the line ${\bf x}= (-2,4,1) + t (5,-1,2)$ and ${\cal P}$ is the plane x + 3y - z = 5.

ii.

Suppose ${\cal L}$ is the line ${\bf x}= (0,2,-3) + t (4,1,-1)$ and ${\cal P}$ is the plane x - y + 2z = 2.

2.

Suppose ${\cal L}_1$ and ${\cal L}_2$ are distinct lines in ${\bf R}^{3}$.

(a)

What are the possible geometric relationships between these lines? How would you determine the geometric relationship?

(b)

When would the two lines lie in a plane? How would you determine the plane which contains both lines?

(c)

What is the relationship between each of the following pairs of lines?

i.

The line through (1,2,1) with direction vector (1,1,-1) and the line through (-1,0,3) with direction vector (-1,2,1).

ii.

The line ${\bf x}= (-1,3,2) + t(2,1,0)$ and the line ${\bf x}= (2,1,4) + t(1,0,2)$.

iii.

The line ${\bf x}= (0,2,-1) + t(2,-2,3)$ and the line ${\bf x}= (1,-2,4) + t(-3,3,4.5)$.

3.

(a)

Given three points in ${\bf R}^{3}$, how would you determine if the points are collinear?

(b)

Given four points in ${\bf R}^{3}$, how would you determine if the points are coplanar, that is, if there is a plane containing all four points?

(c)

For each of the following sets of points, determine if they are collinear or coplanar. If so, find the line or plane which contains them.

i.

P = (2,1,0), Q = (1,3,-1), R = (2,4,-3).

ii.

P = (1,2,-3), Q = (2,0,-1), R = (-1,6,-7).

iii.

P = (2,1,0), Q = (1,3,-1), R = (2,4,-3), S = (1,-1,3).

iv.

P = (1,2,-3), Q = (2,0,-1), R = (-1,6,-7), S = (0,2,2).

startsection section10mm-.5 Discussion 5 Introduction to Optics

 

Optics is the study of the behavior of light, the visible and near visible parts of the electromagnetic spectrum, as it moves through an optical system. The optical system might be a single mirror or prism, a zoom lens on a camera, a laser, or the human eye. Our goal in this discussion will be to apply the language of vectors to develop several fundamental concepts in optics. In the next discussion, we will use the computer to create and analyze models of more complicated optical systems that use the ideas we develop here.

In a uniform or homogeneous medium, like a gas or a lens, very narrow beams of light travel in straight lines. In order to visual this, we will think of ``light rays'' passing through a medium. When light strikes a mirror or passes between media of different densities, the rays change direction according to simple geometric rules. We will represent a ray of light by a vector, a displacement vector, indicating the path of light in a medium. The initial point of the vector will be located at the source of the light or where the light ray enters the medium and the endpoint will be located where the light ray leaves the medium or at a ``viewing'' point. (See Figure .)

  

Figure: A schematic diagram of light passing through a prism from point A to point D. The vector $\overline{AB}$ represents the light in air, the vector $\overline{BC}$ represents the light ray in the prism, and the vector $\overline{CD}$ represents the light ray in air.

The speed of light in a homogeneous medium is a constant which depends on the density of the medium, the denser the medium, the lesser the speed of light. The geometric rules for the behavior of light depend on the relative speeds of light in adjacent media rather than the actual speeds. Thus it will turn out to be useful to work with ratio of the speed of light in a vacuum, c, to the speed of light v in a medium. The quotient, c/v is called the Index of refraction of the substance, which we denote by n. The indices of refraction of several common substances are shown in Figure .

  

Figure: Indices of refraction for some common substances.

Fermat's Principle

The geometric rules that govern reflection and refraction can be determined experimentally, but they are also a mathematical consequence of the physical law known known as Fermat's principle, after the mathematician Pierre de Fermat, who formulated the principle in 1657. It states that light traveling between two points follows the path which takes the least amount of time. We can apply this principle to calculate the path that light takes when reflected in a mirror and when it passes from one medium to another. In order to carry out our calculations, it will be helpful to recall the relationship between velocity, distance and time. If light travels from P to Q in a homogeneous medium it follows the straight line segment between the points. If L denotes the distance between them, which is the length of the vector $\overline{PQ}$, and v is the speed of light in the medium, it will take L/v units of time.

    

Figure: (a) A schematic diagram of a light ray reflecting off a plane mirror. (b) A schematic diagram of a light ray passing from a rarer medium to a denser medium.

startsection subsection10mm.5 Exercises

1.

Fermat's Principle for Reflection. In order to apply Fermat's principle to a ray of light that travels from a point P to a point R after being reflected off of a mirror at a point Q, we must find the location Q that minimizes the total time for the path. It will be sufficient to work in the plane. Figure contains a schematic diagram of a light ray reflecting off a mirror. Rather than find the value of x which minimizes time, we will find the angle I₁ and I₂ which minimize time. Note that the angles are measured counterclockwise from the line normal to the mirror at Q. The angle I₁ is called the angle of incidence and I₂ is called the angle of reflection.

(a)

Express the total time it takes to travel along the vectors $\overline{PQ}$ and $\overline{QR}$ as a function of x. (Notice that a, b, and d are constants.)

(b)

Find the sines of the angles I₁ and I₂ as functions of x.

(c)

In order to minimize the total time as a function of x, we must differentiate total time with respect to x and set the result equal to 0. Use this differentiated expression to express $\sin I_{2}$ in terms of $\sin I_{1}$.

(d)

What does this tell you about I₁ and I₂?

2.

Fermat's Principle for Refraction Now let us apply Fermat's principle to a light ray that passes from a homogeneous medium with index of refraction n₁ to a homogeneous medium with index of refraction n₂, so that the speed of light in the first medium is v₁ = c/n₁ and the speed of light in the second medium is v₂ = c/n₂. Figure contains a schematic diagram for a light ray traveling from a point P in the first medium to the interface of the two media at Q and then on to the point R in the second medium. Once again we will find the relationship between the I₁, the angle of incidence and I₂, the angle of refraction, rather than the location of Q.

(a)

Express the total time it takes to travel along the vectors $\overline{PQ}$ and $\overline{QR}$ as a function of x. (Notice that a, b, and d are constants.)

(b)

Find the sines of the angles I₁ and I₂ as functions of x.

(c)

In order to minimize the total time as a function of x, we must differentiate total time with respect to x and set the result equal to 0. Use this differentiated expression to express $\sin I_{2}$ in terms of $\sin I_{1}$.

(d)

What does this tell you about I₁ and I₂?

    

Figure: (a) Figure for Exercise 3. The image of an object located at P is viewed from R. (b) Figure for Exercise 4. The refracted image of an obect at P is viewed from R.

3.

When we perceive the image of an object in a mirror, we know from experience that the image has been reflected and that the object is located on our side of the mirror. Of course, on occasion we can be fooled by a reflected image and we perceive it to be elsewhere than it is. For example, suppose that you are located at the point R in Figure and that you see the reflected image of an object located at P. Thus your eye receives the ray of light traveling from P to Q to R.

(a)

If you were to be ``fooled'' by the image, where would you perceive the object to be located?

(b)

Of course, images that we see in a mirror are reversed. Why is this the case? Use a diagram to explain your answer.

4.

Just as our eye can be fooled by a reflected image, it can be fooled by a refracted image. For example, suppose that you are located at R in in the air and are looking at an object at P in water. Thus your eye receives the ray of light traveling from P to Q to R.

(a)

If you were to be ``fooled'' by the image, where would you perceive the object to be located?

(b)

The size of objects that we see underwater are distorted. How are they distorted and why? Use a diagram to explain your answer.

5.

The relationship that you found between the angle of incidence and angle of refraction in Exercise 2 can be used to give precise answers to questions like those in Exercise 4. Say, for example, that you are looking over the side of a boat on a calm lake at objects under the surface of the water. Use the value 1.00 for the index of refraction of air and 1.33 for the index of refraction of water. (Remember that our observing an object means that a light ray travels from the object to our eye.)

(a)

(See Figure .) Suppose your eye is located two feet above the surface of the water and you look at an object that is three feet in diameter and four feet under the surface of the water. How large does the object appear to you from your vantage point?

(b)

(See Figure .) Suppose that a small object is located two feet under the surface of the water a lateral distance of four feet from your location. Determine the angle between your line of sight and the vertical segment $\overline{PQ}$ when you look at the object. How deep under the water does the object appear to be?

(c)

(Part (b) continued.) Suppose you move your head closer to the water (along $\overline{PQ}$) and the object remains located at R. What happens to the angle between your line of sight and $\overline{PQ}$ when you look at the object? In particular, what happens when your eye approaches the surface of the water? Explain your answer.

    

Figure: (a) Figure for Exercise 5(a). (b) Figure for Exercise 5(b),(c).