Vectors provide an alternative way to represent a translation. A vector has a direction and a length—the exact features that are involved in moving a point in a translation.
Points to Consider
Think about some special transformation vectors. Can you picture what each one does to a figure in a coordinate plane?
This lesson was about two-dimensional space represented by a coordinate grid. But we know there are more than two dimensions. The real world is actually multidimensional. Vectors are well suited to describe motion in that world. What would transformation vectors look like there?
Review Questions
Prove that any translation in the coordinate plane is an isometry. Given: A translation moves any point horizontally and vertically.
Let and be points in the coordinate plane.
Prove:
[Hint: Express and in terms of and .]
A triangle has vertices , and . The vector for a translation is .
[Example 3] How does this transformation move the original triangle?
Write a transformation vector that would move the solid figure to the red figure.
The transformation vector moves a point to the point . What are the coordinates of the original point?
Write a transformation vector that would move the point to the origin.
How does the translation vector move a figure?
The point is moved by the translation vector . How far does the original point move?
Write a translation vector that will move all points .
Point is moved by the translation vector . Fill in the blanks. (____,____).
Review Answers
The triangle moves up .
The figure does not move; it says the same.
along the vector.
) or
Matrices
Learning Objectives
Use the language of matrices.
Add matrices.
Apply matrices to translations.
Introduction
A matrix is a way to express multidimensional data easily and concisely. Matrices (plural of matrix) have their own language. They also have their own arithmetic. You will learn some of the basic matrix terms, as well as how to add two matrices, and how to tell if that is even possible for two matrices!
Matrices have many applications beyond geometry, but in this lesson we’ll see how to use matrices to perform a translation in a coordinate plane.
Matrix Basics
Simply stated, a matrix is an array (or arrangement) of numbers in rows and columns. Brackets are usually used to indicate a matrix.
is a matrix (read "two by three matrix").
It has rows and columns. These are the dimensions of the matrix.
The numbers in the matrix are called the elements.
An element is located according to its place in the rows and columns; is the element in row , column .
Matrices can represent real-world information.
Example 1
A company has two warehouses in their Eastern Region, where they store three models of their product. A matrix can represent the numbers of each model available in each warehouse.
Here the row number of an element represents the warehouse number, and the column number represents the model number.
There are items of Model in Warehouse .
There are items of Model in Warehouse .
How many items are in Warehouse altogether?
Use row . There are Model Model , and Model in Warehouse .
There are items in Warehouse .
How many Model items are there altogether?
Use column . There are in Warehouse and in Warehouse .
There are Model items in all.
Matrices in the Coordinate Plane
Matrices can represent points in a coordinate plane.
A matrix can represent the coordinates of the vertices of a polygon.
is is is
Each row represents the coordinates of one of the vertices.
Example 2
What are the coordinates of point in the image above?
The coordinate of point is ; the coordinate is .
Matrix Addition
Matrices have their own version of arithmetic. To add two matrices, add the elements in corresponding positions in the matrices.
Add the elements in row column . Add the elements in row column . And so on.
Place each sum in the corresponding position in a new matrix, which is the sum of the original two matrices.
The sum matrix has the same dimensions as the matrices being added.
To add two matrices, they must have the same dimensions.
Example 3
The company in example 1 has a western region as well as an eastern region. The western region also has two warehouses where they store the three models of their product. The matrices represent the numbers of each model available in each warehouse by region.
To prepare a report, Stuart adds the two matrices. This will give him the combined information for both regions.
a) What does the mean in the sum matrix?
is the total number of Model items in both Warehouse buildings in both regions.
b) How many Model items are there in all?
There are Model items: in both Warehouse buildings and in both Warehouse buildings.
c) How many items—any model, any location—are there in all?
. This is the sum of all the elements in the sum matrix:
Translations
You worked with translations in the coordinate plane earlier.
A translation moves each point a horizontal distance and a vertical distance .
The image of point is the point .
Matrix addition is one way to represent a translation.
Recall the triangle in example 2.
is , is is
Each row in the matrix below represents the coordinates of one of the vertices.
In the translation, suppose that each point will move to the right and down.
is , is is .
Each of original points , , and moves to the right, and down.
The coordinates of the image of any point will be the point . The translation can be represented as a matrix sum.
Example 4
What is the image of point in this translation?
The second row represents point . The image of is .
Notice:
The rows of the second matrix are all the same. This is because each point of the triangle, or any point, moves the same distance and direction in this translation.
If the translation had moved each point to the left and up, then the second matrix in the sum would have been:
Lesson Summary
A matrix is an arrangement of numbers in rows and columns. Matrices have their own brand of arithmetic. So far you have learned how to add two matrices.
Matrices have many applications, for example in business and industry. One use of matrices is in working with transformations of points and figures in a coordinate plane. In this lesson you saw that addition of matrices can represent a translation. An unusual feature of a translation matrix is that all the rows are the same.
Points to Consider
In upcoming lessons we’ll learn about two kinds of multiplication with matrices. We’ll then use multiplication of matrices to represent other types of transformations in the coordinate plane, starting with reflections in the next lesson.
Something you rely on all the time, but probably don't think about very much is the fact that any two real numbers can be added and multiplied and the result is also a real number. Matrices are different from real numbers because there are special conditions for adding and multiplying matrices. For example, not all matrices can be added because in order to add two matrices the addends must have the same dimensions. The conditions on matrix multiplication are even more interesting, as you shall see shortly.
Review Questions
Fill in the b
lanks.
Matrix below represents the vertices of a triangle. How does the triangle move in the translation represented by matrix
Matrices and are defined below.
Matrix represents the vertices of a triangle. How does the triangle move in the translation represented by the matrix sum?
Write a matrix so that
A translation moves square to square .
Write a matrix, , for the vertices of square
Write a matrix, , for the vertices of square
Write a translation matrix for which
Review Answers
Does not move
right
left and down
left
up
left and up
Reflections
Learning Objectives
Find the reflection of a point in a line on a coordinate plane.
Multiply matrices.
Apply matrix multiplication to reflections.
Verify that a reflection is an isometry.
Introduction
You studied translations earlier, and saw that matrix addition can be used to represent a translation in a coordinate plane. You also learned that a translation is an isometry.
In this lesson, we will analyze reflections in the same way. This time we will use a new operation, matrix multiplication, to represent a reflection in a coordinate plane. We will see that reflections, like translations, are isometries.
You will have an opportunity to discover one surprising—or even shocking!—fact of matrix arithmetic.
Reflection in a Line
A reflection in a line is as if the line were a mirror.
An object reflects in the mirror, and we see the image of the object.
The image is the same distance behind the mirror as the object is in front of the mirror.
The “line of sight” from the object to the mirror is perpendicular to the mirror itself.
The “line of sight” from the image to the mirror is also perpendicular to the mirror.
Technology Note - Geometry Software
Use your geometry software to experiment with reflections.
Try this.
Draw a line.
Draw a triangle.
Reflect the triangle in the line.
Look at your results.
Repeat with different lines, and figures other than a triangle.
Now try this.
Draw a line
Draw a point.
Reflect the point in the line.
Connect the point and its reflection with a segment.
Measure the distance of the original point to the line, and the distance of the reflected point to the line.
Measure the angle formed by the original line and the segment connecting the original point and its reflection.
Let’s put this information in more precise terms.
Reflection of a Point in a Line:
Point is the reflection of point in line if and only if line is the perpendicular bisector of .
Reflections in Special Lines
In a coordinate plane there are some “special” lines for which it is relatively easy to create reflections.
the axis
the axis
the line (this line makes a angle between the and axes)
We can develop simple formulas for reflections in these lines.
Let be a point in the coordinate plane.
We now have these reflections of :
Reflection of in the axis is .
[coordinate stays the same, coordinate becomes opposite]
Reflection of in the axis is .
[coordinate becomes opposite, coordinate stays the same]
Reflection of in the line is .
[switch the and coordinates]
A look is enough to convince us of the first two reflections. We’ll prove the third one.
Example 1
Prove that the reflection of point in the line is the point .
Here is an “outline” proof.
First, we know the slope of
Slope of is .
Next, let's assume investigate the slope of .
Slope of is .
Therefore, we have just shown that and are perpendicular.
is perpendicular to (product of slopes is ).
Finally, we can show that is the perpendicular bisector of .
Midpoint of is .
Midpoint of is on ( and coordinates of are the same).
is the perpendicular bisector of .
Conclusion: and are reflections in the line .
Example 2
Point is reflected in the line . The image is . is then reflected in the axis. The image is . What are the coordinates of ?
We find one reflection at a time.
Reflect in is .
Reflect in the axis. is .
Reflections Are Isometries
A reflection in a line is an isometry. Distance between points is “preserved” (stays the same).
We will verify the isometry for reflection in the axis. The story is very similar for reflection in the axis. You can write a proof that reflection in is an isometry in the Lesson Exercises.
The diagram below shows and its reflection in the axis, .
Conclusion: When a segment is reflected in the axis, the image segment has the same length as the original segment. This is the meaning of isometry. You can see that a similar argument would apply to reflection in any line.
Matrix Multiplication
Multiplying matrices is a little more complicated than addition. Matrix multiplication is sometimes called a “row-by-column” operation. Let’s begin with examples.
Let and
Notice:
The product is a matrix.
The number of rows in the product matrix is the same as the number of rows in the left matrix being multiplied.
The number of columns in the product matrix is the same as the number of columns in the right matrix being multiplied.
The number of columns in the left matrix is the same as the number of rows in the right matrix.
To compute a given element of the product matrix, we multiply each element of that row in the left matrix by the corresponding element in that column in the right matrix, and add these products.
Some of this information can be stated easily in symbols.
If is an matrix, then must be an matrix in order to find the product .
is an matrix.
Let’s look again at matrices and above:
We found . Is true? Surprisingly, we cannot even calculate . This would have us multiplying a left matrix that is times a right matrix that is . This does not satisfy the requirements stated above. It’s not that does not equal the fact is, does not even exist! Conclusion: Multiplication of matrices is not commutative.
Translated loosely, some matrices you can’t even multiply, and for some matrices that you can multiply, the operation is not commutative.
Example 3
Do the following operation:
Notice: This multiplication in effect adds the elements of each row of the left matrix for the first element in the product matrix, and inserts a for the second element in each row of the product matrix.
Matrix Multiplication and Reflections
We know from earlier work how reflections in the axis, the axis, and the line affect the coordinates of a point. Those results are summarized in the following diagram.
Now we can use matrix arithmetic to express reflections.
Given a point in the coordinate plane, we will use matrix multiplication to reflect the point. Note: In all the matrix multiplications that follow, we multiply on the left by a reflection matrix on the right. Remember (from above), left and right placement matter!
Reflection in the axis: Multiply any point or polygon matrix by
.
Proof.
Reflection in the axis: Multiply any point or polygon matrix by
.
P
roof.
Reflection in : Multiply any point or polygon matrix by
.
Proof. The proof is available in the Lesson Exercises.
Example 4
The trapezoid below is reflected in the line .
What are the coordinates of the vertices of the image of the trapezoid?
1. Write a polygon matrix for the coordinates of the vertices of the trapezoid.
2. Multiply the polygon matrix by (on the right).
3. Interpret the product matrix.
The vertices of the image of the trapezoid are , and .
Lesson Summary
A point or set of points, such as a polygon, can be reflected in a line. In this lesson we focused on reflections in three important lines: the axis, the axis, and the line .
CK-12 Geometry Page 43