BELLWORK 1 A hot air balloon is being held in place by two people holding ropes and standing 35 feet apart. The angle formed between the ground and the rope held by each person is 40. Determine the length of each rope to the nearest tenth of a foot. 22.8 feet
VECTORS IN THE PLANE
LT 7.1 I can represent a vector geometrically.
INTRODUCTION TO VECTORS Temperature, distance, height, area & volume single real number represents its magnitude Scalar Area of a garden 25 ft 2 Pool temperature 78 Height of flagpole 45 ft.
INTRODUCTION TO VECTORS force, velocity and acceleration have magnitude and direction possible directions in a plane are infinite, BUT we only need two numbers, one for magnitude one for direction velocity of a football force of pushing an object wind velocity
EXAMPLE 1: State whether each quantity is a vector quantity or a scalar quantity. Justify your answer. A boat traveling at 15 miles per hour. Scalar quantity: magnitude of 15 miles per hour, but no direction is given.
EXAMPLE 1: State whether each quantity is a vector quantity or a scalar quantity. Justify your answer. A hiker walking 25 paces due west. Vector quantity: magnitude of 25 paces, direction due west
EXAMPLE 1: State whether each quantity is a vector quantity or a scalar quantity. Justify your answer. A person s weight on a bathroom scale. Vector quantity: magnitude = person s mass, direction downward pull due to gravity.
EXAMPLE 2: THINK PAIR SHARE State whether each quantity is a vector quantity or a scalar quantity. Justify your answer. A parachutist falling straight down at 12.5 mph. Vector quantity: magnitude = 12.5mph, direction is down.
EXAMPLE 2: THINK PAIR SHARE State whether each quantity is a vector quantity or a scalar quantity. Justify your answer. A child pulling a sled with a force of 40 newtons. Scalar quantity: magnitude = 40 newtons, direction is not given.
EXAMPLE 2: THINK PAIR SHARE State whether each quantity is a vector quantity or a scalar quantity. Justify your answer. A car traveling 60 mph east of south. Vector quantity: magnitude = 60mph, direction is east of south.
DIRECTIONS QUADRANT BEARINGS S35 E A B C D E F G 30 W of N N30 W 10 W of N N10 W 15 E of N E15 N
DIRECTIONS TRUE BEARINGS True bearings are directional measurements where the angle is measured clockwise from north. Three digits are used to describe a true bearing. For example 5 is written as 005. The bearing of Karen to Stephen can be described as 063.
A VECTOR geometrically a directed line segment tip terminal point initial point
A VECTOR geometrically on the coordinate plane. The directed line segment from A to B
A VECTOR geometrically in standard position Standard Position Ø Initial point (tail) at the origin Ø Terminal point (tip) at a point (a, b)
A two dimensional vector in standard position is an ordered pair of real numbers, denoted in component form as a, b 2,3
The length (scalar) of the vector represents its magnitude, denoted v. Magnitude can represent distance, speed or force. The direction of the vector is the directed angle between the vector and the positive x-axis.
EXAMPLE 3: USE A RULER AND A PROTRACTOR TO DRAW AN ARROW DIAGRAM FOR EACH QUANTITY. INCLUDE A SCALE. a = 20 feet per second at a bearing of 030.
EXAMPLE 3: USE A RULER AND A PROTRACTOR TO DRAW AN ARROW DIAGRAM FOR EACH QUANTITY. INCLUDE A SCALE. v = 75 pounds of force at 140 to the horizontal.
EXAMPLE 3: USE A RULER AND A PROTRACTOR TO DRAW AN ARROW DIAGRAM FOR EACH QUANTITY. INCLUDE A SCALE. z = 30 miles per hour at a bearing of S60 W to the horizontal.
EXAMPLE 3: YOU TRY! USE A RULER AND A PROTRACTOR TO DRAW AN ARROW DIAGRAM FOR EACH QUANTITY. INCLUDE A SCALE. a) t=20 feet per second at a bearing of 065 b) u=15 miles per hour at a bearing of S25 E c) m=60 pounds of force at 80 to the horizontal
TYPES OF VECTORS: Parallel Vectors: Have the same or opposite direction but not necessarily the same magnitude. Equivalent Vectors: Have the same magnitude and direction. Opposite Vectors: Have the same magnitude but opposite direction. a -a
ADDING VECTORS GEOMETRICALLY When two or more vectors are added, their sum is a single vector called the resultant. Method 1:Triangle method: Tip-to-Tail 1) Translate b so that the tail of b touches the tip of a. a 2) The resultant is the vector from the tail of a to the tip of b. b
ADDING VECTORS GEOMETRICALLY Method 2:Paralleogram method: Tail-to-Tail 1) Translate b so that the tail of b touches the tail of a. 2) Complete the parallelogram that has a and b as two of its sides. 3) The resultant is the vector that forms the diagonal of the parallelogram. a b
EXAMPLE 4: FIND THE RESULTANT OF TWO VECTORS. In an orienteering competition, Tia walks N50 E for 120 feet and then walks 80 feet due east. How far and at what quadrant bearing is Tia from her starting point? 185 feet at a bearing of N66 E
EXAMPLE 5: EXPLORE WITH YOUR PARTNER! a) Add 2 parallel vectors with the same direction. b) Add 2 parallel vectors with opposite directions. c) Add 2 opposite vectors.
SCALAR MULTIPLICATION WITH VECTORS If a vector v is multiplied by a scalar k, the scalar multiple kv has a magnitude of k v. Its direction is determined by the sign of k. u If k > 0, kv has the same direction as v. u If k < 0, kv has the opposite direction as v.
EXAMPLE 6: Draw a vector diagram of 3x ¾ y. y x
LT 7.2 - I can solve vector problems and resolve vectors into their rectangular components
USE VECTORS TO SOLVE NAVIGATION PROBLEMS. Vector addition and trigonometry can be used to solve vector problems involving triangles which are often oblique. In navigation, a heading is the direction in which a vessel is steered to overcome other forces, such as wind or current. The relative velocity of the vessel is the resultant when the heading velocity and other forces are combined.
EXAMPLE 7: An airplane is flying with an airspeed of 310 knots on a heading of 050. If a 78-knot wind is blowing from a true heading of 125, determine the speed and direction of the plane relative to the ground. 1) Draw a diagram to represent the heading and wind velocities. 2) Use the Law of Cosines to find g, the planes speed relative to the ground. 3) Find the heading of the resultant (θ), use Law of Sines.
Resolving a vector Two or more vectors whose sum is a vector r are called components of r. Components can have any direction, however it is often useful to resolve a vector into two perpendicular components.
The force r exerted to pull the wagon can be thought of as the sum of a horizontal component force x that moves the wagon forward and a vertical component force y that pulls the wagon upward.
Example 8: Heather is pushing the handle of a lawnmower with a force of 450 newtons at an angle of 56 with the ground. a) Draw a diagram that shows the resolution of the force that Heather exerts into its rectangular components.
Example 8: Heather is pushing the handle of a lawnmower with a force of 450 newtons at an angle of 56 with the ground. b) Find the magnitudes of the horizontal and vertical components of the force.