Vectors Scalars Vectors Magnitude only Time, temperature, speed Magnitude and Direction Forces, position, velocity Need two pieces of info to complete!

Forces are Vectors Two 100-lb forces act on a point Does the total force = 200 lb? If the forces caused the point to move, which way would it go? x y F1 = 100 lbs Need more information: Direction of each line 35° F2 = 100 lbs 60°

Vector Addition Given two vectors, A and B Notation: R = A + B Add vectors head to tail B + A = A + B A R B A R B

Trig Laws Sum of interior angles = 180° Sine law: Cosine law: A B C    if A = B,  =  Cosine law: if  = 90°, A2 = B2 + C2

Vector Addition x y F1 = 100 lbs 85° a 35° F2 = 100 lbs 60° R

Vector Subtraction S = C – D = C + (-D) S C -D D

Vector Operations Multiplication by Scalar Direction doesn’t change Magnitude increases 2.5A A -0.8A

Resolving Vectors One vector can be broken into 2 others FR = F1 + F2 You can find two pieces of information Given F1, find magnitude, direction of F2 Given directions of F1,F2, find both magnitudes Given magnitudes of F1,F2, find both directions Draw out what you know Use trig laws

Resolving Vectors Given FR and F1, find magnitude, direction of F2 F1

Resolving Vectors Given FR and the directions of F1 & F2, find both magnitudes F1 F2 FR

Cartesian Vectors Note that all 2-D vectors can be resolved into 2 perpendicular vectors x y F1 = 100 lbs 35° Fy Fx

Adding Cartesian Vectors Resolve all vectors into x and y components Rx = S(All x components) Ry = S(All y components) R = Rx + Ry |R| = (Rx2 + Ry2)1/2 a = tan-1(Ry/ Rx)

Adding Cartesian Vectors x y F1 = 100 lbs F2 = 100 lbs 35° 60°