Mathematical Investigations IV
Name:
Vectors
Getting To the Point
Rotations
In MI 2, we studied transformations by multiplying a 2x2 matrix by a 2x1 coordinate
matrix. At this point, we can view that coordinate matrix as a vector. Let's review some of that
 2 1 3


work. Reminder: On the TI-89, to enter the matrix  0 4 1 , on the home screen, enter:
 5 2 3


[ 2, -1, 3; 0, 4, 1; 5, -2, 3 ]. That is, begin and end the matrix with square brackets. Separate the
rows with semi-colons. You may wish to give your matrix a name by storing it. Alternatively,
you may use Apps > Data/Matrix Editor, and choose Matrix to enter a matrix.
 a b
2
2
In this worksheet, we want to study matrices of the form 
 , where a  b  1 .
b a 
Let v = 1, 0 . For the moment, place this vector with its tail at the origin. (Some of these will
be easier to do by hand.)
1.
0 1 1 
Find 
   . Make a quick sketch. What does this transformation do to v?
 1 0  0 
2.
 1 0  1 
Find 
   . Make a quick sketch. What does this transformation do to v?
 0 1 0 
Vectors 7.1
Rev. F08
Mathematical Investigations IV
Name:
3.


a. Find 



1
2
3
2
 3  1 

2    . Make a quick sketch. What does this transformation do to v?
 
1  
 0 
2 
 1  3

 2 
2
2

   . Sketch v and w. Does the
b. Now let v = 2,6 and find w =
 
 3
1  

 6 
 2
2 
transformation have the same effect as it did above?
c. Find the angle between v and w using a dot product. Does your answer agree with the
work above?
Vectors 7.2
Rev. F08
Mathematical Investigations IV
Name:
4.
5.
 2  2 

 2 
2
2
   . Sketch v and the resultant vector w. What does this
Find w = 
 2  2  

  6 
 2
2 
transformation do to v? Confirm this by using the dot product.
 3 4   

 2 
5
5

Find w = 
. Sketch v and the resultant vector w. What does this
 4 3   
 5 5   6 
transformation do to v? Confirm this by using the dot product.
Vectors 7.3
Rev. F08
Mathematical Investigations IV
Name:
6.
Generalize: The transformation matrix that will rotate a vector counterclockwise by 
radians is:
How is this consistent with the original premise that a 2  b2  1 ?
7.
8.
a. Find a matrix that rotates vectors by
30° in a counterclockwise direction.
b. Find a matrix that rotates vectors by 45°
in a clockwise direction.
Enter your transformation matrix from problem 6 above into your calculator, using , and
give it a name.
Let  = /12. (Store it on your calculator by entering: /12  .) Now multiply your
transformation matrix by v = 2,6 . State the result. (Lots of square roots!)
Vectors 7.4
Rev. F08