Chapter 2.1 Vectors and Translation
We interact with our environment by transforming it and we study objects in our environment by studying how they change under various transformations.
To understand a thing, we might try to view it as some transformed version of something else that we already understand.
This chapter of our study on the principles of calculus will primarily focus on the rigid motions of the plane, the transformations that do not change distances.
The principle of transformation is to study concepts by studying how the simplest examples of the concepts transform into the more complicated examples.
These transformations are the translations, rotations, and reflections of the plane.
We will also study scaling and inversion.
We will learn how to develop precise mathematical descriptions of operations and functions based on our intuition about these operations that come from our innate understanding of space and quantity.
Abstract Translations of the Plane
Concepts in mathematics come directly from our experiences. These concepts are just translating real world ideas into a mathematical language that allows us to do many things.
In this lecture, we will formalize the notion of translation in the plane.
For motivation, imagine that you are quickly showing someone next to you where something is across a room. What do you do? You point to the thing.
You create an arrow, with your finger being the head of your arrow and your shoulder being the tail.
View an arrow as acting on a point that sits at the tail of the arrow. The arrow moves the point from the tail of the arrow to the head of the arrow.
You can think of the arrow as a slide and the point as a person. Then the action of the arrow to the point is simply the person going down the slide.
This basic idea can actually be seen in real number addition.
Example 1
Rather than view addition of real numbers as an operation that takes two real numbers to a third, view addition as taking a real number to an arrow.
The sum \(a+b\) may be thought of as an arrow \(a+\) acting to move the point \(b\) to a new point.
It is not really a point that adds to another point in the line, it is an arrow that moves points in the line. The difference between points is an arrow.
Example 2
View \(2+\) as an arrow and draw a picture to geometrically represent \(2+5\).
The expression \(2+5\) can be interpreted as take the number \(5\) and move it to the right (\(+2\)) to the point \(7\).
Arrows similarly move points in the plane. There is just more freedom in the choice of arrows that move points.
Example 3
Where does the arrow below move the given point below?
The head of the arrow indicates where the point moves to.
This is how we mathematically define an arrow by using the coordinates of the points.
Difference of two Points
Given points \(p=(p_1,p_2)\) and \(q=(q_1,q_2)\) in the plane, define their difference to be the arrow that translates \(p\) to \(q\).
Denote this difference by \(q - p\) which is given by the formula \[(q_1-p_1,q_2-p_2).\]
Let’s look at some examples.
Example 4
What is \((3,5) - (1, 2)\)? Compare this with the arrow defined by \((6,8) - (4, 5)\).
Calculate to gett \[(3,5)-(1,2)=(2,3)\] and \[(6,8) - (4, 5)=(2,3).\] 
Ultimately, the arrow does the same thing despite the difference in location.
A single arrow only moves a single point. It is not the arrow that is important, it is how the arrow moves a point that is important. Here is a mathematical definition that captures this idea.
Equivalent
For any arrows \(A_1\) and \(A_2\) with tails respectively at \(p_1\) and \(p_2\), if
\(A_1\) changes the \(x\)-coordinate of \(p_1\) by the same amount that \(A_2\) changes the \(x\)-coordinate of \(p_2\), and if
\(A_1\) changes the \(y\)-coordinate of \(p_1\) by the same amount that \(A_2\) changes the \(y\)-coordinate of \(p_2\),
then \(A_1\) and \(A_2\) are equivalent.
We can now define the mathematical concept of vector.
Vector
A vector is the set of all arrows that are equivalent to a particular arrow.
A vector \(v\) acts on a point \(p\) as the arrow in \(v\) whose tail is at \(p\).
Vectors can be added together and addition is defined by the way in which vectors move points.
The vector \(v_2 +v_1\) moves the point \(p\) to \(v_1+p\) and then moves this point by \(v_2\).
Vector addition is both associative and commutative.
Here we define both the zero vector, the vector that keeps points stationary, and the additive inverse of a vector.
Zero Vector and Additive Inverse of a Vector
Denote by \(0\) the vector that acts on the plane by translating every point in the plane to itself.
Let \(-v\) be the vector such that \[v + (-v)= 0.\]
Vectors and the Method of Coordinates on a Plane
Vectors can be characterized by how they move the point \((0,0)\). So we can write out a vector in the following way.
Vector In Coordinates
If the vector \(v\) moves \((0,0)\) to \((a, b)\), then denote \(v\) by \(\langle a, b\rangle\).
This is an expression for \(v\) “in coordinates.”
Here is a specific example.
Example 5
The vector \(V=\langle 2, 7\rangle\) moves the point \((0,0)\) to the point \((2, 7)\).
In this next example, let’s practice visualizing vectors
Example 6
Compute \(V - W\) where \[V = \langle 5,2\rangle \quad\text{and} \quad W = \langle 2, 0\rangle.\] What does this vector look like? Compute \((V-W)+(1,3)\) and visualize this on a graph.
Calculate to get \(V - W=\langle 3,2\rangle\) and \((V-W)+(1,3)=(4,5).\) We can visualize this below.
The expression \((V-W)+(1,3)\) tells us that the point \((1,3)\) is translated in the direction of \((V-W)\).
Here are some additional facts about vectors.
Vector Addition
A vector adds to another vector in the following way: \[\langle a, b\rangle + \langle c, d\rangle = \langle a +c, b+d\rangle.\]
A vector translates a point in the following way: \[\langle a, b\rangle + (c,d) = (a+c, b+d).\]
The difference of points is a vector: \[(c,d) - (a,b) = \langle c-a, b-d\rangle.\]
This gives us the rather nice notation: \[((c,d) - (a,b)) + (a,b) = \langle c-a, d-b\rangle + (a,b) = (c,d).\]
Important question: Why do we write the coordinates for a vector as \(\langle a, b\rangle\) rather than \((a,b)\)?
Because the vector is not a point in the plane, it acts by moving points in the plane.
Example 7
The vector \(V\) moves the point \((2,5)\) to the point \((3, 1)\). Calculate \(V + (1,8)\)
Calculate \(V\) by computing \((3, 1) - (2,5)\); therefore \(V=\langle 1,-4 \rangle.\)
Confirm this is correct by making sure that \(V+(2,5)\) leads results in the point \((3,1)\).
\[V+(2,5)=\langle 1,-4\rangle +(2,5)=(3,1).\] Calculate \(V+(1,8)\) to get \(( 2, 4 )\).
Translating Sets and Graphs
If \(V\) is a vector, then \(V\) acts pointwise on any subset of the plane. What this means is it will move all points in the subset in the direction of \(V\).
Translation
Namely, if \(A\) is a subset of the plane, define \(V+A\) by \[V + A = \{V+(x,y)\colon (x,y)\in A\}.\]
A function \(T\colon \mathbb R^2 \to \mathbb R^2\) is a translation if there is a vector \(V\) so that for every \(p\) in \(\mathbb R^2\), \[T(p) = V+p.\]
Here is an example of the above definition.
Example 8
Let \(V\) be the vector below and let \(A\) be the set below. Draw the set \(V + A\).
The expression \(V+A\) means \(A\) is translated by \(V=\langle 4,2\rangle\)
Because a function is just a subset of the plane, vectors can act on functions to produce new functions.
If a function \(f\) is given by a formula, we can determine the formula that describes \(V+f\).
Start with a function, \(f\), with \[f = \{(x,f(x))\colon x\in D(f)\}.\]
Shift \(f\) by adding to each point the vector \(V=\langle h, k\rangle\) to obtain the function \[g = V+f = \{(x+h,f(x)+k)\colon x\in D(f)\}.\]
The resulting function, \(g\), is given by the equation \[g(x) = f(x-h) + k,\] which has domain equal to \(\mathcal D(f) +h\) and range equal to \(\mathcal R(f) + k\).
Graphical differences include:
Shifts the graph right by \(h\): \(y = f(x-h)\).
Shifts the graph up by \(k\): \(y = f(x) + k\).
We give the details of the argument below.
Example 9
Explain why \(g(x) = f(x-h) + k\) is a shift right by \(h\), shift up by \(k\) and has domain equal to \(\mathcal D(f) +h\) and range equal to \(\mathcal R(f) + k\).
The function \(f\) is a set:
\[f=\{(x,f(x))\colon x \in D(f)\}.\] It is all ordered pairs \((x,f(x))\) where \(x\) is in the domain of \(f\).
Take \(V=\langle h,k\rangle.\)
Then \[\begin{align*}V+f&=\{(x+h,f(x)+k)\colon x\in D(f)\}&&\text{Set }X=x+h \text{ or } X-h=x\\&=\{(X,f(X-h)+k)\colon X-h\in D(f)\}\end{align*}.\]
Hence \(g=\{(x,f(x-h)+k)\colon x-h \in D(f)\},\) so a formula for \(g\) is \(g(x)=f(x-h)+k\).
Visualize in the following way:
which means \(D(g)=D(f)+h\) and \(R(g)=R(f)+k\).
Now let’s look at a specific example.
Example 10
Sketch \(\langle 1,3\rangle+{\rm pow}_2\) and find an equation describing the points in this new function
Recall that \({\rm pow_2} =x^2\) and is a parabola. So \(\langle 1,3\rangle+{\rm pow}_2\) takes all points \((x,x^2)\) and translates them in the direction \(\langle 1,3\rangle.\)
An equation that the describes the new function is \[y=f(x-1)+3\] or \[y=(x-1)^2+3.\]
We will revisit an idea we’ve learned before but we reinterpret it using vectors.
Parallel
The lines \(L_1\) and \(L_2\) are parallel if there is a vector \(V\) such that \[V+L_1 = L_2.\]
If \(L_1\) and \(L_2\) are not vertical then they are parallel if and only if they have the same slope.
Let’s look at an example in which we use vectors to answer the question.
Example 11
Take \(L\) to be the line that passes through \((2,4)\) and \((6,7)\). Find an equation of a line parallel to \(L\) that passes through \((3, 9)\)
The line \(L\) passes through \((2,4)\) and \((6,7)\) so it has a slope of \(m=\frac{7-4}{6-2}=\frac{3}{4}\).
One example of a line parallel to \(L\) is \(f(x)=\frac{3}{4}x\). However, this line doesn’t pass through \((3,9)\). Translate it so it does.
\(\langle 3,9\rangle +f =\{(x,f(x-3)+9)\colon x\in \mathbb{R}\}.\)
The final line is \(y=\frac{3}{4}(x-3)+9\). This is parallel to \(L\) and crosses at \((3,9)\).
In conclusion, there are different ways to approach a concept and vectors can be used to describe things we’ve talked about before.