08 Plane vectors · g10-08-01 · 5–10 min
Move the arrow without changing the vector
Must equal vectors start at the same point?
Predict: Move an arrow of displacement(3,2) to a different starting point. Predict its length and components.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Start x = 0; Start y = 0; Displacement x = 3; Displacement y = 2 | |
Case 2
Start x = 2; Start y = -1; Displacement x = 3; Displacement y = 2 | |
Case 3
Start x = 2; Start y = -1; Displacement x = 0; Displacement y = 0 | |
Revise: My claim needs the condition… because…
Exit: Are arrows from(0,0) to(2,−1) and(3,4) to(5,3) equal?
08 Plane vectors · g10-08-02 · 5–10 min
A detour and its final displacement
Does changing the order of two displacements change the destination?
Predict: Compare a+b andb+a for a=(2,1),b=(−1,3).
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
a₁ = 2; a₂ = 1; b₁ = -1; b₂ = 3 | |
Case 2
a₁ = 2; a₂ = 1; b₁ = -1; b₂ = 3 | |
Case 3
a₁ = 2; a₂ = 1; b₁ = -2; b₂ = -1 | |
Revise: My claim needs the condition… because…
Exit: Find(2,1)−(−1,3).
08 Plane vectors · g10-08-03 · 5–10 min
A negative scalar reverses direction
What does multiplication by−2 change?
Predict: Compare a,2a,−2a and0a for a=(2,1).
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
a₁ = 2; a₂ = 1; Scalar λ = 2 | |
Case 2
a₁ = 2; a₂ = 1; Scalar λ = -2 | |
Case 3
a₁ = 2; a₂ = 1; Scalar λ = 0 | |
Revise: My claim needs the condition… because…
Exit: Find−3(1,−2) and its magnitude.
08 Plane vectors · g10-08-04 · 5–10 min
Two directions need to be independent
Can any pair of vectors locate every point uniquely?
Predict: Keep e₁=(1,0). Compare e₂=(0,1) ande₂=(1,0) for target(3,2).
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
e₁ x = 1; e₁ y = 0; e₂ x = 0; e₂ y = 1; Target x = 3; Target y = 2 | |
Case 2
e₁ x = 1; e₁ y = 1; e₂ x = -1; e₂ y = 1; Target x = 3; Target y = 2 | |
Case 3
e₁ x = 1; e₁ y = 0; e₂ x = 2; e₂ y = 0; Target x = 3; Target y = 2 | |
Revise: My claim needs the condition… because…
Exit: Do(1,0) and(2,0) form a plane basis?
08 Plane vectors · g10-08-05 · 5–10 min
What does the sign of a dot product tell us?
Does a dot product of0 always mean a right angle?
Predict: Compare two nonzero vectors at60°,90°,120°, then let one length become0.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
|a| = 3; |b| = 2; Angle (degrees) = 60 | |
Case 2
|a| = 3; |b| = 2; Angle (degrees) = 90 | |
Case 3
|a| = 0; |b| = 2; Angle (degrees) = 90 | |
Revise: My claim needs the condition… because…
Exit: What extra condition is needed in “a·b=0 implies a⊥b”?
08 Plane vectors · g10-08-06 · 5–10 min
Translate a midpoint and a distance
What changes when both endpoints receive the same translation?
Predict: Translate A=(−1,0),B=(3,2) by(2,1); predict the new midpoint and distance.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
A x = -1; A y = 0; B x = 3; B y = 2; Translation x = 0; Translation y = 0 | |
Case 2
A x = -1; A y = 0; B x = 3; B y = 2; Translation x = 2; Translation y = 1 | |
Case 3
A x = 1; A y = 1; B x = 1; B y = 1; Translation x = 2; Translation y = 1 | |
Revise: My claim needs the condition… because…
Exit: Find the midpoint of(−1,0),(3,2), then translate it by(2,1).
