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

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”

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

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

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

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

Using Trig Functions to Find Vector Components

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

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| = 1 Figure  Example: Vector A in x-y plane. Components Ax, Ay: A  Axi + Ayj Figure 

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

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

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

Example

Example

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

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