Slide1 :
Analytic Method of Addition : Analytic Method of Addition Resolution of vectors into components:
YOU MUST KNOW &
UNDERSTAND
TRIGONOMETERY TO
UNDERSTAND THIS!!!!
Analytic Method : Analytic Method 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”
Slide4 : V is resolved into components: Vx & Vy
V Vx + Vy (Vx || x axis, Vy || y axis)
Brief Trig Review : 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
Slide6 : Define trig functions in terms of h, a, o:
Signs of sine, cosine, tangent : Signs of sine, cosine, tangent Trig identity: tan(θ) = sin(θ)/cos(θ)
Using Trig Functions to Find Vector Components : Using Trig Functions to Find Vector Components
Example : Example V = displacement 500 m, 30º N of E
Example : Example Consider 2 vectors, V1 & V2. Want V = V1 + V2
Example 3-2 : Example 3-2
Problem Solving : Problem Solving You cannot solve a vector problem
without drawing a diagram!
Example 3-3 : Example 3-3
Alternate Analytic Method : Alternate Analytic Method Laws of Sines & Law of Cosines from trig.
Appendix A-7, p A-8, 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
Slide15 : 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