1. Two-Dimensional Motion and Vectors
Vector Operations

2. Coordinate System in Two Dimensions
Can change the orientation of the system so that motion is along an axis
Or can apply a coordinate system along two axes
(4,-5)

3. Determining Resultant Magnitude and Direction
Use Pythagorean theorem to find magnitude when vectors form a right triangle
(length of one leg)2 + (length of other leg)2 = (length of hypotenuse)2
a2+b2=c2
C equals displacement, velocity, acceleration, etc
Use inverse tangent function to find angle of the resultant
This is the direction
angle = inverse tangent of (opposite leg)/(adjacent leg)
θ=tan-1(opp/adj)

4. Determining Resultant Magnitude and Direction
An archaeologist climbs the Great Pyramid in Giza, Egypt. The pyramid’s height is 136m and it’s width is 2.30*102m. What is the magnitude and the direction of the displacement of the archaeologist after she has climbed from the bottom of the pyramid to the top?
h = 136m w = 2.30*102m / 2 = 115m

5. Determining Resultant Magnitude and Direction
r2 = h2 + w2 = (115)2 = = 31721
r = 178m
θ = tan-1(opposite/adjacent) = tan-1(height/width) = tan-1(136/115) = 49.8°

6. Resolving Vectors into Components
Components of a vector – the projections of a vector along the axes of a coordinate system
Use sine function to find opposite leg
opposite leg = hypotenuse * (sine of angle)
opp = hyp * (sinθ)
Use cosine function to find adjacent leg
adjacent leg = hypotenuse * (cosine of angle)
adj = hyp * (cosθ)

7. Resolving Vectors into Components
Find the components of the velocity of a helicopter traveling 95km/h at an angle of 35° to the ground.
95km/h km/h
35° y 35° x

8. Resolving Vectors into Components
opp = hyp * (sinθ)
Vertical = hypotenuse * (sinθ)
Vertical = 95(sin35) = 54 km/h
adj = hyp * (cosθ)
Horizontal = hypotenuse * (cosθ)
Horizontal = 95(cos35) = 78 km/h

9. Adding Vectors That Are Not Perpendicular
Form right triangles from each leg of the vector
Use sine and cosine to get total x and y displacements
Use Pythagorean theorem and inverse tangent to get magnitude and direction of resultant

10. Adding Vectors That Are Not Perpendicular
A hiker walks 27.0km from her base camp at 35° South of East. The next day, she walks 41.0km in a direction 65° North of East and discovers a forest ranger’s tower. Find the magnitude and direction of her resultant displacement between the base camp and the tower.

11. Adding Vectors That Are Not PerpendicularR 35
27.0km 65
41.0km

12. Adding Vectors That Are Not Perpendicular
27km
y1=hyp. cos(55)
=(27)cos(55)
=(-15 km)
x1=hyp. sin(55)
=(27)sin(55)
=22km y1 x1 55

13. Adding Vectors That Are Not Perpendicular
y2=hyp. sin(65)
=(41)sin(65)
=37km
x2=hyp. cos(65)
=(41)cos(65)
=17km
41.0km 65 X2 Y2

14. Adding Vectors That Are Not PerpendicularR Yt Xt