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Vectors in Two and Three Dimensions

Part 1: Arithmetic of Vectors in the Plane

A vector in the xy-plane may be thought of as an arrow connecting the origin (0,0) to any point (a,b). For example, the following picture shows the vector connecting to the point (10,10) in green and the vector connecting to the point (-3,4) in red.

  1. Enter the commands in your worksheet to draw your own version of the picture above.

When v is the vector from (0,0) to (a,b), we write v = (a,b). That is, we identify the arrow with the point at its head. Now why, you may ask, would we go to the trouble of introducing another word to describe points in the plane? Well, the notation v = (a,b) is just a description of the vector that happens to look like a description of a point -- but there is much more involved in the mathematical concept of vector.

First, each such arrow has both a length and a direction. We already know that length and direction are a pair of numbers that can represent a point in the plane -- that's what polar coordinates do. So, in a sense, the vector concept incorporates both rectangular and polar coordinates simultaneously. We use absolute value bars to denote the length of a vector. Thus, if v = (a,b), then .

  1. Find the length and direction of each of the vectors in your figure in step 1.

  2. Change the definitions of the two vectors in your worksheet, and repeat steps 1 and 2.

  3. Think about the connection between rectangular and polar coordinates, and write down a list of at least 12 vectors that describe points on the unit circle.

  4. Vectors of length 1 -- called unit vectors -- are sometimes useful for indicating directions. Explain why the vector is a unit vector in the direction indicated by the polar angle .

  5. For each of the following unit vectors, find the corresponding polar angle .
    (a)           (b)   
    (c)           (d)   

Second, unlike points in the plane, vectors have an arithmetic. In particular, vectors can be added and subtracted just by adding or subtracting their coordinates:

(a,b) + (c,d) = (a + c, b + d)

(a,b) - (c,d) = (a - c, b - d)

In the next several steps, we will see how to visualize these operations geometrically.

  1. On a sheet of paper, write down a pair of vectors -- such as the most recent pair u and v in your worksheet -- and add them. Make a sketch on your paper that shows each of the two vectors, as well as their sum, u + v. (If you have graph paper handy, that might be useful, but any paper will do.) What do you notice?

  2. Use the applet below to replicate and confirm your paper sketch. In the applet, the green and yellow vectors are the vectors to be added, and the red vector is the sum. You can move the yellow and green vectors to their approximate positions by clicking on them and dragging with the mouse button down. Since there are no scales on the axes, you will have to choose your own approximate scales. [Hint: The yellow and green vectors are restricted in their lengths -- try positioning one of them as your longer vector first, and then scaling the other vector accordingly.]

The grey lines in the applet complete a parallelogram with two sides determined by the yellow and green vectors, and the sum is the diagonal of the parallelogram. The grey lines also suggest that it might be useful to free the vectors from the constraint of having their tails tied down at the origin. Specifically, we say that any arrow with the same length and direction as a given arrow from the origin is the same vector as the given one. Thus, the grey side opposite the green arrow is the green arrow (when given the proper direction), and the grey side opposite the yellow arrow is the yellow arrow. When viewed that way, we see that the sum (red arrow) is the third side of a triangle formed when two sides are the yellow and green arrows -- in either order -- with the tail of the second arrow tied to the head of the first.

  1. On your sheet of paper, write down a pair of vectors -- the same u and v will do -- and subtract them. Make a sketch on your paper that shows each of the two vectors, as well as their difference, u - v. What do you notice?
  2. Click the "Difference" button in the applet above, and confirm your paper sketch. Note that the applet is using the freedom to move a vector to a parallel position in its representation of the difference by the red arrow.
  3. What happens if you move the tail of the difference vector to the origin -- as you probably did in your paper sketch? Does the difference vector agree with the coordinates you computed? How does u - v correspond to u + (-v), where -v means negating the coordinates of v?
  4. Explain why u - v is the vector that can be added to v to get u as the sum.

Third (remember first and second?), a vector can be multiplied by a number ("scaled") to lengthen or shorten it -- and, in the case of a negative multiplier, to reverse its direction. This process is called scalar multiplication, and it is defined by the formula

c= c(a, b) = (ca, cb)

for a vector = (a,b) and a scalar c. Since two vectors are parallel if and only if they point in the same or opposite direction, parallel vectors must be scalar multiples of each other. [Note: Numbers are also called scalars because of their role in scaling vectors -- thus, "scalar" is both an adjective and a noun.]

  1. Are the vectors (1,2) and (2,3) parallel? Why or why not? Are the vectors (2,4) and (-1,-2) parallel? Explain.

To summarize our "second" and "third" observations about vectors, we have seen that vectors can be combined and manipulated using the vector operations of addition, subtraction, and scalar multiplication.

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