Adding Vectors Head to Tail: A Lesson Plan

To add two vectors, draw the second one starting where the first one ends: head to tail. The sum, or resultant, goes from the start of the first to the end of the second. In components it is even simpler: add the x parts and the y parts. (3, 1) + (1, 3) = (4, 4).

Learning objectives

• Add vectors graphically, head to tail.
• Add vectors by components and find the magnitude with Pythagoras.
• Use vector addition for displacements and velocities.

The opening question

"You walk 3 km east, then 4 km north. How far are you from where you started?" Students say 7 km; the answer is 5 km. Distances walked add; displacements add as vectors.

The lesson, click by click

1. Click 0: vector a = (3, 1) from the origin.
2. Click 1: vector b = (1, 3) moves so that its tail sits on the head of a.
3. Click 2: the resultant a + b, from the origin to (4, 4).
4. Click 3: components: (3 + 1, 1 + 3) = (4, 4), and magnitude √(4² + 4²) ≈ 5.66.
5. Click 4, a river: a boat heads across at 4 m/s while the current pushes it at 3 m/s: it moves at 5 m/s, about 37° downstream.

Common mistakes to watch for

• Adding magnitudes. |a| + |b| ≈ 3.16 + 3.16 = 6.32, but |a + b| ≈ 5.66. They are equal only for vectors in the same direction.
• Tail to tail. Drawing both from the same point and joining their heads gives the difference, not the sum.
• Mixing components. Add x with x and y with y, never x with y.

Check questions

1. (2, −1) + (−5, 3)? ((−3, 2).)
2. Magnitude of (6, 8)? (10.)
3. Two forces of 3 N east and 4 N north act on a box. The resultant? (5 N, about 53° north of east.)

Building it in Tracé

1. Add a Vector with x component 3 and y component 1, and a second with 1 and 3.
2. On click 1, move the second vector so its tail is at (3, 1): it slides into place.
3. Add the resultant on click 2, in a different colour, and the component calculation on click 3.
4. For forces on an object, the Free-body diagram object draws several arrows from one box.

Questions

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