Vectors Questions Vectors vs. Scalars Vectors – Graphical addition Vectors – Components addition Example 3-2. Example 3-3. Quiz next week (10-15 min)

Vectors and Scalar Quantities Vector quantities Position Velocity Acceleration Force Momentum Scalar quantities Time Mass Energy Temperature (Dr. Hart’s dog)

Vector Addition – why? Multiple-leg trip Boat velocity + stream velocity Multiple forces on object Accelerated circular motion Momentum Electric Forces Magnetic Forces

Vectors – Graphical addition Method 1 Sequential movement “A” then “B”. Etown/Lancaster, Mt. Gretna detour. Tail-to-tip method. Right angles, non-right angles. Method 2 Simultaneous little-bit “A” and little bit “B” Velocity, paddling across the current Force, pulling a little in x and a little in y Parallelogram method. Right angles, non-right angles Two methods are equivalent C B A C B A

Vectors – Graphical subtraction If C = A + B Then B = C - A B = C + -A C B A -A B C

Vectors - components If C = A + B (vector C is sum of vectors A and B) Then C = Cx + Cy ( C can be broken into components Cx and Cy) Method 3 - Break all vectors into components, add components, reassemble result C B A Cy C Cx

Vectors – Components addition Components - Simplest method Break into x and y components Add components like accounting sheet Recombine to form final vector Trig definitions Break up - right triangle, sin, cos, tan Combine - Pythagorean theorem, arctan Example 3-2 Example 3-3

Trigonometry For any right triangle Basic trig functions 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒=ℎ𝑦𝑝 ∙ sin 𝜃 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡=ℎ𝑦𝑝 ∙ 𝑐𝑜𝑠 𝜃

Mail Carrier Displacement

Mail carrier displacement X-component Y-component D1 22 km D2 + 47 km cos(60) - 47 km sin(60) D + 23.5 km -18.7 km Insert sign by inspection 𝐷= 𝐷 𝑥 2 + 𝐷 𝑦 2 =30 𝑘𝑚 𝑡𝑎𝑛𝜃= 𝐷 𝑦 𝐷 𝑥 =−.796 𝜃=−38.5° (𝑏𝑒𝑙𝑜𝑤+𝑥 𝑎𝑥𝑖𝑠)

Plane Trip

Plane Trip 𝐷= 𝐷 𝑥 2 + 𝐷 𝑦 2 =960 𝑘𝑚 𝑡𝑎𝑛𝜃= 𝐷 𝑦 𝐷 𝑥 −1.25 x-component y-component D1 620 km 0 km D2 +440 km cos(45) +311 -440 km sin45 -311 D3 -550 km cos(53) -330 -550 km sin53 -439 D 601 km -750 𝐷= 𝐷 𝑥 2 + 𝐷 𝑦 2 =960 𝑘𝑚 𝑡𝑎𝑛𝜃= 𝐷 𝑦 𝐷 𝑥 −1.25 𝜃=−51° (𝑏𝑒𝑙𝑜𝑤+𝑥 𝑎𝑥𝑖𝑠)

Other examples Problem 9 Problem 10 Problems 6-8