1. 2-D Motion: Vector Properties
Trigonometric Methods

2. Vector Properties
What is one disadvantage of adding vectors by the graphical method?
Is there an easier way to add vectors?

3. TRIG!!!!! Pythagorean Theorem For Right Triangles c2 = a2 + b2
(length of hyp)2 = (length of one leg)2 + (length of one leg)2

4. TRIG!!!!! Other Trig Properties: (soh cah toa) sin(θ) = opp/hyp
cos(θ) = adj/hyp
tan(θ) = opp/adj

5. Vector Addition - Sample Problems
12 km east + 9 km east = ?
Resultant: 21 km east
12 km east + 9 km west = ?
Resultant: 3 km east
12 km east + 9 km south = ?
Resultant: 15 km at 37° south of east
12 km east + 8 km north = ?
Resultant: 14 km at 34° north of east
For the first two items, have students predict the answer before showing it. They generally have no trouble with these two problems. Point out that the process is the same if it is km/h or m/s2. Only the units change. These problems do not require trigonometry because the vectors are in the same direction (or opposite directions).
For the third problem, most students will probably remember the Pythagorean theorem and get the magnitude, but many will fail to get the direction or will just write southeast.
Show students how to use the trig identities to determine the angle. Then, explain why it is south of east and not east of south by showing what each direction would look like on an x-y axis. If they draw the 9 km south first and then add the 12 km east, they will get an answer of 53° east of south (which is the same direction as 37° south of east). After your demonstration, have students solve the fourth problem on their own, and then check their answers. Review the solution to this problem also.
Insist that students place arrows on every vector drawn. When they just draw lines, they often draw the resultant in the wrong direction.
You might find the PHet web site helpful ( If you go to the Math simulations, you will find a Vector Addition (flash version). You can download these simulations so your access to the internet is not an issue. You can show both the resultant and components using this simulation. Students could also use this at home to check their solutions to problems.

6. Resolving Vectors into Components
Opposite of vector addition
Vectors are resolved into x and y components
For the vector shown at right, find the vector components vx (velocity in the x direction) and vy (velocity in the y direction).
Answers:
vx = 89 km/h
vy = 32 km/h
Review the first solution with students, and then let them solve for the second component.

7. Adding Non-Perpendicular Vectors
Four steps
Resolve each vector into x and y components
Add the x components (xtotal = x1 + x2)
Add the y components (ytotal = y1 + y2)
Combine the x and y totals as perpendicular vectors
Explain the four steps using the diagram.
Show students that d1 can be resolved into x1 and y1 . Similarly for d2. Then, the resultant of d1 and d2 (dashed line labeled d) is the same as the resultant of the 4 components.

8. Practice Problem
A camper walks 4.5 km at 45° north of east and then walks 4.5 km due south. Find the camper’s total displacement.
Answer
3.4 km at 22° S of E

9. Practice Problem
A pirate walks 50 m at 35° S of W, thinking he is on the right path to his buried treasure.
He then remembers he had to rebury it and changes course. He now walks 75 m at 80° N of E. Find his total displacement.
Answer: m at 59.1° N of W