Vector Spaces, I
Life in Two and Three Dimensions

A point in the plane can be represented mathematically by a pair of real numbers as shown in the picture below.

A pair of real numbers is called a vector in R² and is written

or, sometimes,

Vectors in R² can represent many things besides points in the plane. For example, they can represent --

• An arrow starting at the origin and ending at the point v = (x, y) as shown in the figure below.
  

• Motion in the plane starting at an arbitrary point and moving x units horizontally and y units vertically as shown in the movie below.
  Motion in the plane

  When we think of a vector v = (x, y) as representing motion we often picture it as an arrow starting at an arbitrary point and ending at the point that would be reached after moving x units horizontally and y units vertically. All of the arrows in the figure below represent the same motion.

  

• A bundle of two different things. For example, the vector (3, 4) might represent a bundle with 3 pounds of sugar and 4 pounds of flour.

• Velocity: For example, the velocity of an object traveling from southwest toward the northeast at a speed of 60 miles per hour is represented by the vector
  

• Force: For example, if a wind blowing from southwest toward the northeast exerts a force of 60 pounds on a person then the force can be represented by the vector
  

• Vector addition: If x = (x₁, x₂) and y = (y₁, y₂) then
  x + y = (x₁ + y₁, x₂ + y₂)

• Scalar multiplication: If x = (x₁, x₂) and c is a real number then
  c x = (c x₁, c x₂).

Do each of the calculations below by hand and then check your answer in your CAS window.

1. (1, 2) + (5, -3)

2. (5, -3) + (1, 2)

3. 4 (2, 4)

4. 2 (1, 2) + 3 (4, 1)

5. 2 (u, v) + 3 (x, y)

6. t (1, 2) + s (3, 5)

7. s (u, v) + t (x, y)

If we think of a vector as representing an arrow starting at the origin then we can visualize vector addition as shown in the figure below. We begin with two vectors a (blue) and b (green) representing two sides of a parallelogram with one vertex at the origin. The vector a + b represents the diagonal line shown in red.

• One geometric definition of a parallelogram is that in each pair of two opposite sides the sides are parallel. Show that the four sides of the figure whose vertices are represented by the origin, the vector x, the vector y, and the vector x + y have this property.

• One geometric definition of a parallelogram is that in each pair of two opposite sides the sides have the same length. Show that the four sides of the figure whose vertices are represented by the origin, the vector x, the vector y, and the vector x + y have this property.

Scalar multiplication stretches or compresses a vector as shown in the figure below. The graph on the left shows a vector x in red. The middle graph shows the vectors 2 x in green and -2 x in blue. The graph on the right shows the vectors (1/2) x in green and -(1/2) x in blue.

These two operations have some very nice properties called vector space properties. Click on the icon below for a list of these properties and some of their implications.

Using these vector space operations we can describe translations very cleanly.

Notice that this formula uses vectors in two different ways -- the vector x represents a point in the plane and the vector a represents motion in the plane.

We can also define a new family of operations on the plane, called dilations, that magnify or reduce an image.

Notice

• If |a| > 1 then the dilation magnifies the image.
• If |a| < 1 then the dilation reduces the image.
• If a is negative then the dilation also rotates the image by pi radians or 180 degrees.

The figure below shows an original image in black and then the result of applying dilations with

• a = 2 (red).
• a = 1/2 (blue)
• a = -2 (green)

We can prove some interesting facts about translations and dilations from the vector space properties.

1. Prove that any two translations commute -- that is, if S and T are translations then

   S(T(x)) = T(S(x)).

   answer

2. Prove that any two dilations commute -- that is, if S and T are dilations then

   S(T(x)) = T(S(x)).

   answer

If x and y are vectors representing two points then y - x is the vector representing the motion required to go from x to y. That is,

For a proof see the mathematical infrastructure module on vector spaces.

Develop formulas to answer each of the questions below and check your formulas by working out a few examples in your CAS window.

1. Find the point that is halfway between x and y. answer

2. Find the point that is on the line segment between x and y but is twice as far from x as it is from y. answer

3. Find the point that is on the line segment between x and y but is twice as far from y as it is from x. answer

4. Find the point that is on the line determined by the two points x and y but is beyond y and is twice as far from x as it is from y. answer

5. Find the point that is on the line determined by the two points x and y but is beyond x and is twice as far from y as it is from x. answer

Generalizing from the exercises above we can see that any point of the form P = a + t (b - a) = (1 - t) a + t b is on the line determined by the two points a and b. If 0 <= t <= 1 then P is between a and b. If t < 0 then P is beyond a. If 1 < t then P is beyond b. The Java applet at the right illustrates this. The point represented by the vector a is indicated by a red dot and the point represented by the vector b is represented by a blue dot. If you click anyplace along the line determined by these two points, the corresponding value of t will appear in the "t" box. You can also type a value of t in the t box and click on the move button. Then the corresponding point will be indicated on the line determined by a and b.

• Consider a triangle whose three vertices are represented by the vectors A, B, and C. Show that any point, P, inside or on this triangle can be written in the form
  P = a A + b B + c C

  where

  a + b + c = 1

  and

  0 <= a <= 1
  0 <= b <= 1
  0 <= c <= 1

• Show that any point, P, in the plane can be written in the form
  P = a A + b B + c C

  where

  a + b + c = 1.

• In the preceding problem you established a connection between the geometric ideas of a triangle and an arbitrary point in the plane and the algebraic idea of vectors of the form
  P = a A + b B + c C

  where

  a + b + c = 1.

  Explore this connection. What does it mean geometrically if one of the coefficients a, b, or c is zero? What does it mean geometrically if two of them are zero?

• What does it mean geometrically if one of the coefficients a, b, or c is negative? What does it mean geometrically if two of them are negative?

The applet on the right illustrates the last three problems above. Notice that there is a large triangle with a colored dot at each of its three vertices. Suppose the vertices are represented by the vectors A, B, and C. Click any place on the applet. The point, P, at which you clicked can be written in the form P = a A + b B + c C where a + b + c = 1. Notice at the bottom of the applet the value of each of these three coefficients is printed. Test your work above by clicking at various points to see what happens to the coefficients.

Movies or animations are created by displaying a sequence of frames of still images one-after-another to produce the appearance of motion. Broadcast television (in the United States), for example, displays 30 complete frames per second.

Suppose, for example, that we want to move an image two units right and three units upward over a period of three seconds. We will use 91 frames, numbered -- 0, 1, 2, ... 90. To create the i-th frame we apply the translation

to each point in the original image. Notice that when i = 0 we get the original image unchanged and when i = 90 the original image is translated point-by-point by (2, 3) as desired.

Use your CAS window to help do the following problems and to check your answers.

1. The left side of the figure below shows two triangles -- a blue triangle formed by connecting the points (-4, 4), (-3, -2), and (-2, -4) by straight lines -- and a red triangle formed by translating the blue triangle by (6, 6). Suppose that you want to move the blue triangle smoothly to the position of the red triangle using a total of 31 frames numbered -- 0, 1, 2, ... 30. How would you generate each frame? The right side of the figure below shows all the frames superimposed on each other.
   

2. Suppose that you want to rotate the blue triangle shown in the figure below counterclockwise around the origin through a full circle in four seconds using a total of 121 frames. How would you generate each frame? The same figure shows frames 30, 60, and 90 in red.
   

3. The center of the blue triangle shown in the figure below is at the point (0, 1.5) Suppose that you want to rotate the triangle around the origin through a full circle counterclockwise keeping it upright in four seconds using a total of 121 frames. How would you generate each frame? The same figure shows frames 30, 60, and 90 in red.
   

4. Suppose you want to create the appearance of an image growing by a factor of three over a period of five seconds using a total of 151 frames. How would you generate each frame?

In the three-dimensional world in which we live the location of a point is given by three numbers or coordinates, (x, y, z) as shown in the figure below.

• The first coordinate, x, describes the position of the point along an axis that is perpendicular to the screen -- positive values of x are in front of the screen, toward the viewer, and negative values of x are behind the screen.

• The second coordinate, y, describes the position of the point along a horizontal axis -- positive values of y are to the viewer's right and negative values of y are to the viewer's left.

• The third coordinate, z, describes the position of the point along a vertical axis -- positive values of z are upward and negative values of z are downward. Most mathematicians label the axes as described above but this labeling is not chiseled in stone. In fact, many people who work with computer graphics use a different labeling -- with the x-axis where our y-axis is; the y-axis where our z-axis is; and the z-axis coming out of the screen toward the viewer.

  

  A vector in R³ is a triple v = (x, y, z) of real numbers. We sometimes use the notation --

  x = (x₁, x₂, x₃)

  Vectors in R³ can represent many things besides points in the three dimensional world. For example, they can represent --

  • An arrow starting at the origin and ending at the point v = (x, y, z) .

  • Motion in the three dimensional world starting at an arbitrary point and moving x units toward or away from the viewer, y units to the left or the right, and z units up or down.

  • A bundle of three different things. For example, the vector (3, 4, 5) might represent a bundle with 3 pounds of sugar, 4 pounds of flour, and 5 pounds of salt.

  • Force: For example, the force exerted by gravity on an object weighing 10 pounds is represented by the vector (0, 0, -10). Notice that the only nonzero coordinate of this vector is the third one and that it is negative because the force of gravity is pulling the object downward.

  

  Just as in R², we define two arithmetic operations on vectors in R³ -- vector addition and scalar multiplication by a real number as follows.

  • Vector addition: If x = (x₁, x₂, x₃) and y = (y₁, y₂, y₃) then
    x + y = (x₁ + y₁, x₂ + y₂, x₃ + y₃)

  • Scalar multiplication: If x = (x₁, x₂, x₃) and c is a real number then
    c x = (c x₁, c x₂, c x₃)

  

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  Do each of the calculations below by hand and then check your answer in your CAS window.

  1. (1, 2, 3) + (5, -3, -2)

  2. (5, -3, -2) + (1, 2, 3)

  3. 4 (2, 4, 6)

  4. 2 (1, 2, 3) + 3 (4, 1, 7)

  5. 2 (u, v, w) + 3 (x, y, z)

  6. t (1, 2, 3) + s (3, 5, 7)

  7. s (u, v, w) + t (x, y z)

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  These two operations have the usual very nice properties called vector space properties. Click on the icon below for a list of these properties and some of their implications.

  

  The exact same formula

  P = t u + (1 - t) v

  that we used to describe points on the line determined by two points u and v in R² works in exactly the same way in R³.

  

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  1. Find the point that is halfway between the two points (1, 2, 3) and (4, 3, 3).

  2. Find the point that is between the two points (1, 2, 3) and (4, 3, 3) but is twice as far from (1, 2, 3) as it is from (4, 3, 3).

  3. Find all the points that are on the line determined by the two points (2, 1, 6) and (6, 2, 4) and are twice as far from (2, 1, 6) as from (6, 2, 4).

  4. Is the point (2, 3) on the line determined by the points (-1, -1) and (5, 7)? answer

  5. Is the point (3, 4) on the line determined by the two points (-1, -1) and (5, 7)? answer

  6. Is the point (1, 2, 3) on the line determined by the two points (2, 1, 3) and (3, 2, 1)? answer

  7. Is the point (4, 3, 2) on the line determined by the two points (3, 2, 1) and (2, 1, 0)? answer

  8. Find a formula that describes the points on the line that starts at the point u and points in the direction v as shown in the figure below. Note that the vector u represents a point and the vector v represents an arrow that starts at u. answer
     

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  Copyright c 1995 by Frank Wattenberg, Department of Mathematics, Montana State University, Bozeman, MT 59717