HC-10. When Vectors are equal:
This supplement was mostly written by Bob Moses and used at Edison HS in 2007-2008.
{Note to teacher: Use this section as needed. If ready, your class may go directly to the Geometric Real Line.}
Different displacements of the same movement
S1: But what about the movement +2?
S2: Yes. Suppose is a movement of +2 which has direction up and magnitude 2.
M: Do you remember we asked you to draw that movement for subtraction problems?
S3: I do. One subtraction problem had a movement of +2 that began at 3 and ended at +5.
S1: Right. And another began at 5 and ended at 3.
S2: That’s right so 5 – 3 was the first subtraction.
S3: And 3 – 5 was the next subtraction problem.
M: Yes, and each displacement is a movement of +2 that has a specifiedstarting point.
S3: Right, 5 – 3 starts at 3 .
S1: And, 3 – 5 starts at 5 .
M: Yes, we will say that 5 – 3 and 3 – 5 are two different displacements.
S2: What do they have in common?
S1: Well, they both have the same movement.
S3: We said two students could have the same height mark on the class height chart.
M: Good point. We will say that 5 – 3 and 3 – 5 are different displacements, different names for the same movement.
S2: So I can write: (5 – 3) = (3 – 5) .
S1: And what does it mean?
M: A movement can produce many different displacements.
S2: So “5 – 3” and “-3 – -5” are two different displacements.
S1: But we know that (5 – 3) = (3 – 5) .
S2: I get it. It means we have two different displacements for the same movements.
M: That’s our point: A movement can produce many displacements. Do you like that way of thinking about “5 – 3” and “–3 – –5” as the same movement?
S1: I think it’s cool. It makes sense that the displacement from 3 to 5 is the same movement as the displacement from–5 to –3 .
S2: Yes, and the movement has the same direction and the same magnitude in both subtractions.
S3: That’s true, it sure does, but you know what? It has different location coordinates. ∆y1 = 5 – 3, has location coordinates 3 and 5.
S1: On the other hand, ∆y3 = –3 – –5, has location coordinates–5 and –3 .
M: What about the movement between points you have been constructing on the group Height Charts? Do you think we could find equal movement vectors?
S1: I don’t think so because our rulers are not precise.
M: That’s true, for now, let’s consider another procedure.
S2: I have an idea. Suppose we ask whether any other height marks are the same distance apart as Bertho’s and Glad’s height marks.
S3: How would that tell us about equal movement vectors?
S1: I get it. Think about the movement BA, where Bertho’s mark is the point A and…..
S2: Don’t tell me, I know: Glad’s mark is the point B.
S3: I get it now. Then we have a displacementvector BA that tells us the comparison of Glad’s mark to Bertho’s mark.
M: You all are really thinking well about this. What are you going to look at next?
S1: I know what. We can take a piece of card stock or paper and lay it along the line of the height chart and mark on it Glad’s mark and Berto’s mark. Then we can usethis to look for other pairs of marks on the height chart that “match up” with Bertho’s and Glad’s marks.
S2: Yes. That’s it! And if we find such points we can label these height marks with “C” and “D”. Then …
S3: Then the displacement vector DC will be equal to the displacement vector BA: DC=BA.
S1: Or possibly, in the other direction: CD=BA.
S2: It depends on whether C is on the same side of D as A is of B.
M: That’s terrific. I think you should use this process to go on a hunt for equal vectors:
HC-10a. Worksheet to hunt for equal vectors on the group height charts:
First choose two height marks and label them: A = ______B = ______
Next, use sheet of paper or card stock with these marks marked on it. Lay it along the height mark line and search for other pairs of height marks that “match up” with points A and B.
If you find one more such pair put A = C and B = D, Record the two names: C = ______D = ______
If you find more than one such pair, use E,F and G,H etc. as your labels.
Write your equations stating the equality of the vectors BA and DC in Abstract Vector Symbols.
______
Write a paragraph explaining:
- What you have done.
- How you did it
- About the relationship between the height marks for A and B and the height marks for C and D. Use the names of the students whose height marks you labeled A, B, C, and D.
HC-10b. Different names for the same location vector
M: Do you remember when we talked about equality between two vectors?
S1: Yes, we said that 2 students may have the same height mark in the Height Chart.
S2: Yes, ______, ______, and ______all have the same height mark on the Class Height Chart.
S3: Yes, so in that case, if we put “A” for Anne’s mark and “B” for Bertha’smark and … .
M: So then what are two representations of their location vectors?
S1: “ yA” is the representation for Anne’s location vector.
S2: and Bertha’s location vector is the symbol yB.
M: So in this case there is only one location vector but it has two names:
S1: Right and we agreed that there was a way to think about the equality of two location vectors.
S2: Right. We write: yA = yB.
S2: We have two different names for the same location vector.
Algebra Project, Inc, 2008. 1