Chapter 4 Pg. 62-79 Vector Addition

4.1: Properties of Vectors Determine graphically the sum of two or more vectors Solve problems of relative velocity Key Terms Graphical representation Algebraic representation Resultant Vector

Representing Vector Quantities Vectors are represented by a line segment with an arrow. They show a magnitude (length) and a direction Displacement vectors are “green” Velocity vectors are “red” This shows a graphical representation of a vector. We use these when drawing vectors diagrams Our lengths have to be to scale to show the magnitude of the vector correctly.

Representing Vector Quantities Algebraic representation Is always in “italicized, boldface print” We may see something for displacement like: d = 50km, SE. This gives both the magnitude and the direction of the vector.

The Resultant Vector Displacement vectors are equal when their displacement and direction are the same. See p. 64 (Figure 4.1, vectors A & B) Even if they do not begin and end at the same point, they can still be considered equal. Unequal vectors – have different length or different direction or both

The Resultant Vector Vectors can be moved graphically, this makes it possible to add/subtract them. Displacement is change in position No matter the route you take home, you displacement is always the same. Resultant vector – a vector equal to the sum of 2 or more vectors. Look @ Figure 4-2 on p. 65 of text

Assignment Worksheet Handout Pg. 4 Questions 1-5

Graphical Addition of Vectors 2 pieces of information needed to solve: Magnitude Angle Graphing it: Draw vectors to where they meet at one point to form a triangle of some sort (can be right, acute, obtuse, does not matter) Draw the resultant of the two or more vectors to show the magnitude of the vectors

Magnitude of Resultant Finding the Resultant: Sum of Two Vectors Vectors meet at right angle: Use Pythagorean Theorem R2 = A2 + B2 Vectors meet at greater than 90* angle: Use Law of Cosines R2 = A2 + B2 – 2ABCosΘ Try Practice Problems 1-4 on pg. 67 of textbook.

Subtracting Vectors Multiplying a vector by a scalar changes length but not its direction unless the scalar is negative. This allows for the ability to subtract vectors in the same way we add them. Δv= v2- v1 or v2+ (-v1) If v1 is multiplied by -1, then it changes its direction and –v1 can be added to v2 to calculate Δv

Relative Velocities Vector addition and subtraction can be useful when solving for relative velocities. Relative velocities is a velocity in relation to another object (can be stationary or moving) Ex: You are on a bus traveling 8m/s North, you get up and walk to toward the front of the bus at 3m/s; what is your relative velocity to someone on the street?

Relative Velocities The addition of relative velocity vectors can even be applied to forces in 2 dimensions. Ex: Airplane pilots, do not just aim their plane toward the destination and fly straight there. They must account for airspeed and calculate how far it will cause their path to deviate. Also the velocity of the airplane in relation to the air is important. The resultant of these calculations can give the actual path of the plane in relation to the ground. Do Practice Problems 5-10 on pg. 71 of text

Assignment 4.1 Review Pg. 71 1-4 Due next class

4.2: Components of Vectors Objectives Establish a coordinate system in problems involving vector quantities Use the process of resolution of vectors to find the components of vectors Determine algebraically the sum of 2 or more vectors by adding the components of the vectors

Choosing a Coordinate System Gridwork work for solving vectors No set way to do this, usually whatever makes problem easiest to solve. In vectors using directions on Earth, Y-axis points North and X-axis points west In anything involving elevation, vertical change is the y-axis; horizontal change is the x-axis

Components Choosing a coordinate system allows us to break the vector into its two components: horizontal and vertical. This is called vector resolution. The original vector is equal to the sum of the 2 component vectors Magnitude and sign of the component vectors are simply called components; all calculations are made off of the components.

Components Can use Trig. to solve for components. Soh Cah Toa Sine, Cosine, Tangent Look at Example Problem on pg. 73 of text Do Practice Problems 11-14 on. Pg. 74

Algebraic Addition of Vectors 2 or more vectors may be added, but first we must find the components of each vector; both X and Y components The resultant of the x components we call Rx and it is the sum of Ax + BX + Cx ….. The resultant of the y components is solved in the same manner. The angle of the resultant is determined using: tanΘ = Ry/Rx You can also just use the Tan-1 function on your calculator Give Practice Problems 15-18 a shot

Assignment 4.2 Review Questions 1-4 Due tomorrow.