Vectors

Vectors Two Types of Quantities Scalar – has magnitude only Ex. Time, mass, volume, speed, distance Vector – has both magnitude and direction Ex. Velocity, electric, magnetic fields, force, displacement

Vectors Vectors are named with capital letters with arrows above the letter. Vectors are represented by arrows (ray). Drawn to scale the arrow shows the magnitude (size) and direction. Ex. A = 35km/hr east Ex. B = 70km/hr east Ex. C = 20km/hr at 350 NE

Vectors All vectors have a head and a tail.

Vector Addition (Tail-Head Method) Vector quantities are added graphically by placing the tail of the second to the head of the first. Resultant Vector (R) – the one single vector that is equal to all the vectors combined. Resultant is drawn from tail of first to head of the last vector

Vector Addition – same direction A + B = R A B B A R = A + B

Vector Addition Example: What is the resultant vector of an object if it moved 5 m east, 5 m south, 5 m west and 5 m north?

Vector Addition – Opposite direction A + (-B) = R A B -B A A + (-B) = R -B

Vectors The sum of two or more vectors is called the resultant.

Vectors Vector Addition Triangle (Tail-Head) Method Ex. Find the resultant vector of A and B. A = 5km/hr east B = 7km/hr north Use the Pythagorean Theorem to find the magnitude of the resultant vector. R R = 8.6km/hr northeast B A2 + B2 = C2 A

Check Your Understanding A plane travels with a speed of 80 km/hr due west. Determine the resultant velocity of the plane if it encounters a 60 km/h headwind to the south.

Right Triangles

Example: How long is the base of the 2 Example: How long is the base of the 2.5 m ramp that is elevated 30o from the ground? How high is the ramp?

Vector Resolution Vector A has been resolved into two perpendicular components, Ax (horizontal component) and Ay (vertical component).

Vector Resolution When resolving a vector graphically, first construct the horizontal component (Ax). Then construct the vertical component (Ay). Using right triangle trigonometry, expressions for determining the magnitude of each component can be derived.

Horizontal Component (Ax)

Vertical Component (Ay)

Drawing Directions EX1: 30° E of S EX2: 30° S of E N N W E W E S S

Drawing Directions EX3: 20° N of W EX4: 20° W of N N N W E W E S S

Ex5: A crate is being pulled by cables along a frictionless surface with a force of 500 kN eastward and by another force of 400 KN @ 30o W of N. What is the Resultant Vector acting on the crate?

EX6: You travel 650 meters east EX6: You travel 650 meters east. You then travel 350 meters south, then 500 meters west, 400 meters north and finally 200 meters west. What direction is your final displacement (resultant vector)?

Negative components If you are given a negative component, it is referring to the placement on a quadrant N (-,+) (+,+) W E (-,-) (+,-) S

Negative Components EX7: Draw the resultant vector with angle that is made up of an x component of 5.5 km and a y component of -3.5 km.