2. Analytic Method of Addition
Resolution of vectors into components:
YOU MUST KNOW &
UNDERSTAND
TRIGONOMETERY TO
UNDERSTAND THIS!!!!

3. Vector Components Consider vector V in a plane (say, xy plane)
Can express V in terms of components Vx , Vy
Finding components Vx & Vy is equivalent to finding 2 mutually perpendicular vectors which, when added (with vector addition) will give V.
That is, find Vx & Vy such that
V  Vx + Vy (Vx || x axis, Vy || y axis)
Finding components
 “Resolving into components”

4. V is resolved into components: Vx & Vy
V  Vx + Vy (Vx || x axis, Vy || y axis)

5. Brief Trig Review
Adding vectors in 2 & 3 dimensions using components requires TRIG FUNCTIONS
HOPEFULLY, A REVIEW!!
See also Appendix A!!
Given any angle θ, can construct a right triangle:
Hypotenuse  h, Adjacent side  a, Opposite side  o

6. Define trig functions in terms of h, a, o:

7. Signs of sine, cosine, tangent
Trig identity: tan(θ) = sin(θ)/cos(θ)

8. Using Trig Functions to Find Vector Components

9. Example
V = displacement 500 m, 30º N of E

10. Unit Vectors UNIT VECTOR  a dimensionless vector, length = 1
Convenient to express vector A in terms of it’s components Ax, Ay, Az & UNIT VECTORS along x,y,z axes
UNIT VECTOR  a dimensionless vector, length = 1
Define unit vectors along x,y,z axes:
i along x; j along y; k along z
|i| = |j| = |k| = Figure 
Example: Vector A in x-y plane.
Components Ax, Ay:
A  Axi + Ayj
Figure 

11. Simple Example Position vector r in x-y plane. Components x, y:
r  x i + y j
Figure 

12. Vector Addition Using Unit Vectors
Suppose we want to add two vectors V1 & V2 in x-y plane:
V = V1 + V2
“Recipe”
1. Find x & y components of V1 & V2 (using trig!)
V1 = V1xi + V1yj V2 = V2xi + V2yj
2. x component of V: Vx = V1x + V2x
y component of V: Vy = V1y + V2y
3. So
V = V1 + V2 = (V1x+ V2x)i + (V1y+ V2y)j

13. Example
Consider 2 vectors, V1 & V2. Want V = V1 + V2

14. Example

15. Example

16. Another Analytic Method
Laws of Sines & Law of Cosines from trig.
Appendix B.4, p B.9, arbitrary triangle:
Law of Cosines: c2 = a2 + b2 - 2 a b cos(γ)
Law of Sines: sin(α)/a = sin(β)/b = sin(γ)/c
See also, Example 3.2

17. Add 2 vectors: C = A + B
Law of Cosines:
C2 = A2 + B2 -2 A B cos(γ)
Gives length of resultant C.
Law of Sines:
sin(α)/A = sin(γ)/C, or sin(α) = A sin(γ)/C
Gives angle α