Journey vs Location

Section 1.1 Journey vs Location

In this section we will consider some foundational concepts that we use in multivariable calculus. Our goal is to recognize the notation and understand what it tells us. When we achieve this we will be able to read the more complicated expressions later in the course and recognize their sometimes simple meanings.

Subsection 1.1.1 Understand Location and Direction

First, we consider combining a location and a direction. These activities illustrate concepts of points and vectors, and it will be used to define lines, planes, and space curves.

Activity 1.

Suppose for each problem that the person begins at the corner of 15th and I St. Use a map (e.g., maps.google.com ¹) to identify the resulting locations.

(a)

If Guido walks 2 blocks north, where is he?

(b)

If Guido walks 5 blocks north then 3 blocks east, where is he?

(c)

If Uwe walks 3 blocks east then 5 blocks north, where is he?

(d)

If Guido walks 2 blocks north then 1 block east then 1 block north then 2 blocks east, where is he?

(e)

If Uwe walks 3 blocks east then 3 blocks north, where is he?

(f)

If Wolfgang rides in a helicopter \(3\sqrt{2}\) blocks northeast, where is he?

(g)

How many locations are 3 blocks east and 3 blocks north of 15th and I St?

(h)

How many ways was this location reached in the examples above? Note all of these use only linear travel (no curves).

Activity 2.

Guido’s parents asked him “Where did you go at noon?” Guido’s reply was “I traveled 4 miles north.” Note Guido fails to tell his parents were he was at noon.

(a)

Where was Guido if he started at Elmore and Abbott?

(b)

Where was Guido if he started at Point Woronzof Park?

(c)

Why should his parents be worried if Guido states that he drove a car, and he started at Elmore and Abbott?

(d)

Draw a starting point. Sketch a path that travels 3 units north then 3 units to the east. Sketch the path from the starting point to the ending point. If Guido follows the first path, does he ever travel northeast?

Activity 3.

On another day Guido tells his parents that he traveled from 15th and I St to 13th and I St then to 13th and L St.

(a)

Do his parents know where he was at the three times mentioned?

(b)

Do his parents know which direction he traveled?

Subsection 1.1.2 Terminology and Notation

Definition 1.1.1. Point.

A point specifies a location.

Points are denoted with capital letters and coordinates (Cartesian) and are specified with ordered lists. For example \(A=(1,3)\text{,}\) and \(B=(-4,7,-2)\text{.}\) Typically points are represented graphically as a dot at the specified location.

Definition 1.1.2. Vector.

A vector specifies a direction and a distance traveled in that direction.

Vectors are denoted with lower case letters with a right arrow above it and are specified with ordered lists. For example \(\vec{x}=(1,3)\text{,}\) and \(\vec{y}=(-4,7,-2)\text{.}\) Other notations for vectors include \(\vec{x}=\langle 1,3 \rangle\) and \(\vec{y}=-4\vec{i}+7\vec{j}-2\vec{k}\text{.}\) The latter relies on properties developed in the exercises below and on defining \(\vec{i}=(1,0,0)\text{,}\) \(\vec{j}=(0,1,0)\text{,}\) and \(\vec{i}=(0,0,1)\text{.}\) Typically vectors are represented graphically as an arrow beginning at the origin — unless otherwise specified — and ending the specified number of units away.

Definition 1.1.3. Directed Line Segment.

A directed line segment specifies two locations and the direction traveled from one to the other.

Directed line segments are specified by stating the two points. The direction is from the first point toward the second. They are denoted by an ordered pair of letters —representing the two points — with a right arrow above it. For example \(\stackrel{\longrightarrow}{AC} \) for \(A=(1,3)\) and \(C=(4,-5)\text{.}\) Directed line segments are typically represented graphically by an arrow beginning at the first point and ending at the second point.

Please note that notations for points, vectors, and various linear objects vary from author to author. You must be careful to check each text before assuming meanings.

Subsection 1.1.3 Vector Arithmetic

These exercises provide an opportunity to review vector arithmetic and recognize what the results are of combining (via arithmetic) points, vectors, and directed line segments. We will use these relationships to understand what notations in future problems mean.

Figure 1.1.4. Demonstration of Graphing 3D by Hand

\begin{align*} \vec{x}+\vec{y} = &\\ (x_1,x_2,x_3)+(y_1,y_2,y_3) = &\\ (x_1+y_1,x_2+y_2,x_3+y_3) \end{align*}

Figure 1.1.5. Vector Addition

Exercises Exercises

Exercise Group.

If you have not learned the vector arithmetic used below and it is not immediately obvious, please see the instructor, tutors, or someone else who can show you the details.

1.

\((2,7)+(-3,4)\)

2.

\((1,1,3)+(0,1,9)\)

3.

\(4(1,1,3)\)

4.

\(\vec{v}=(3,7,-2)\) and \(\vec{w}=(5,-2,4).\)

(a)

\(\vec{v}+\vec{w}\)

(b)

\(\vec{v}-\vec{w}\)

(c)

\(3\vec{v}-4\vec{w}\)

Exercise Group.

While points, vectors, and directed line segments have different interpretations, they can be mixed in arithmetic operations. The result of the arithmetic is considered a point, vector, or other linear object based on context.

Use the following points and vectors for the following problems. \(O=(0,0,0),\) \(A=(2,-3,5),\) \(B=(4,-1,-3).\) \(\vec{u}=(4,4,3),\) \(\vec{v}=(6,2,3).\)

5.

Calculate and sketch a graph for each of the following.

(a)

\(\vec{u},\vec{v},\vec{u}+\vec{v}\)

(b)

\(\stackrel{\longrightarrow}{OA}, \stackrel{\longrightarrow}{AB} \)

6.

What is the vector for \(\stackrel{\longrightarrow}{AB}\text{?}\)

7.

What is the vector for \(\stackrel{\longrightarrow}{BA}\text{?}\)

Exercise Group.

Watch the video in Figure 1.1.4 to learn how to sketch 3D points by hand before continuing. Note, sketching in 3D by hand will lead to a better understanding of functions and surfaces. We learn it to understand them rather than because graphing by hand is a useful skill for daily life.

8.

Calculate and sketch a graph for each of the following.

(a)

\(\vec{u},\vec{v},-3\vec{u}+2\vec{v}\)

(b)

\(\stackrel{\longrightarrow}{OA}, \stackrel{\longrightarrow}{AB} \)

9.

What is the vector for \(\stackrel{\longrightarrow}{AB}\text{?}\)

10.

What point is at the end of \(B-2\vec{u}\text{?}\)

Subsection 1.1.4 Points, Vectors, and Functions

The notation \(\R^n\) is used to express the set of all points or vectors with \(n\) entries. \(\R=\R^1.\)

Definition 1.1.6. Vector Valued Function.

A function is vector valued if and only if the codomain (inclusive of the range) is \(\R^n\) with \(n \ge 2.\)

Note that the domain may be \(\R^m\) for any \(m \ge 1.\) Also note that in a cruel twist of fate for calculus students the expression vector valued function is used for functions whose output is vectors and for functions whose output is points. In this class a function with vector outputs will be denoted like a vector (arrow on top) and a function with point outputs will be denoted without the vector notation. For example \(\vec{v}(x,y)=(2x,2y)\) and \(\ell(t)=(0,3)+(1,1)t.\)

Activity 4.

(a)

For \(B=(5,0,-4)\) and \(\vec{m}=(1,-1,2)\) calculate the following. \(B+\vec{m},\) \(B+2\vec{m},\) \(B+3\vec{m},\) \(B-\vec{m}.\)

(b)

Are the objects in the previous problem points or vectors?

(c)

Calculate 5 points for the function \(\ell_1(t)=(-1,3,2)+(1,1,1)t.\)

(d)

Sketch your 5 points.

(e)

Pretend you are very young and do what comes naturally with a set of dots.

(f)

Based on your sketches what type of function is this?

(g)

Sketch \(\ell_2(t)=(1,3,-2)+(1,1,1)t.\)

(h)

Sketch \(\ell_3(t)=(1,3,-2)+(1,1,-2)t.\)

(i)

Compare and contrast \(\ell_1(t)\) and \(\ell_2(t).\)

(j)

Compare and contrast \(\ell_2(t)\) and \(\ell_3(t).\)

(k)

Write a function for a line through the point \((5,2,0)\) with the direction given by the vector \((1,1,1).\)

(l)

Write a function for a line through the points \((5,2,0)\) and \((6,2,3).\)

(m)

Graph the function \(f(t)=(3+t,1+t^2,2+5t).\)

maps.google.com