6.6 Vectors

Direction: 45⁰ Magnitude=5 5 45⁰ Vector: A vector is a quantity that has both magnitude and direction It is customary to represent a vector by using an arrow terminal point A vector is drawn as a directed line segment, the arrow shows the direction. It does not mean it is a ray. 5 45⁰ initial point Direction: 45⁰ Magnitude=5

Direction: 45⁰ Magnitude=5 5 Vector: A vector is a quantity that has both magnitude and direction A vector has no position. These are all the same vector 5 Magnitude=5 Direction: 45⁰

Direction: 45⁰ Magnitude=5 5 The position vector has its initial point at the origin 5 Magnitude=5 Direction: 45⁰

Direction: 45⁰ Magnitude=5 5 The position vector has its initial point at the origin 5 Magnitude=5 Direction: 45⁰

This is the unit vector i

This is the vector 2i

This is the vector 3i

This is the unit vector j

This is the vector 2j

This is the vector 3j

The position vector has its initial point at the origin and can be represented as the sum of component vectors i and j

The position vector has its initial point at the origin and can be represented as the sum of component vectors i and j

This is the position vector for w=2i+3j Since it is difficult to write vectors using bold print by hand we can also represent this vector as

This is the position vector for v=-2i-3j

Draw three representations of the vector u=i-2j on the graph, be sure to make one the position vector.

Draw three representations of the vector u=-2i+j on the graph, be sure to make one the position vector.

Vector v has initial point P and terminal point Q Vector v has initial point P and terminal point Q. Write v in ai+bj form. Subtract the coordinates of the terminal point from the coordinates of the initial point. P Q

Vector v has initial point P and terminal point Q Vector v has initial point P and terminal point Q. Write v in ai+bj form. Q P

Vector v has initial point P and terminal point Q Vector v has initial point P and terminal point Q. Write v in ai+bj form. Q P

Find the magnitude 3 -4

To find the magnitude of a vector -2 -5

To find the magnitude of a vector

Draw three representations of the vector w=3i - 2j Write the vector v with initial point P (-6,1) and terminal point Q (-2,-3) in ai+bj form. Find the magnitude of the following vectors. u= -12i -5j w= -10j

P 750 1-19 odd

You can multiply a vector by a scalar

You can multiply a vector by a scalar. Find

Find 3w -6u

You can add two vectors geometrically by placing the terminal point of one at the initial point of the other. Find The sum is represented by the resultant vector

You can add two vectors algebraically by adding the corresponding components of the two vectors. Find

Add algebraically and graphically

You can subtract vectors by adding the opposite of the second vector

You can subtract two vectors algebraically by subtracting the corresponding components of the two vectors

You can subtract vectors by adding the opposite of the second vector

You can subtract two vectors algebraically by subtracting the corresponding components of the two vectors

Notebook Link

Find Algebraically.

Find Geometrically

Find Geometrically

Show geometrically and algebraically P 751 #21 -24 Geometrically & Algebraically, #25 – 32Algebraically Show geometrically and algebraically Show algebraically

Find the vector v in ai+bj given the magnitude and the angle it makes with the positive x-axis. 4 b a

Find the vector v in ai+bj given the magnitude and the angle it makes with the positive x-axis. 3

Find the vector v in ai+bj given the magnitude and the angle it makes with the positive x-axis. 3 a

Find the vector v in ai+bj given the magnitude and the angle it makes with the positive x-axis. 2

Find the vector v in ai+bj given the magnitude and the angle it makes with the positive x-axis. 4 b a

The jet stream is blowing at 40 mph and the direction N30°E The jet stream is blowing at 40 mph and the direction N30°E. Express its velocity as a vector v in terms of i and j. N 40mph W E S

A child pulls a wagon with a force of 25lbs on the handle that makes an angle 40° with the ground. Express its velocity as a vector v in terms of i and j. 25lbs

Find the magnitude , to the nearest hundredth and the direction angle θ, to the nearest tenth of a degree . 10 6

Find the magnitude , to the nearest hundredth and the direction angle θ, to the nearest tenth of a degree . 7 -2

Find the magnitude , to the nearest hundredth and the direction angle θ, to the nearest tenth of a degree . 5 -10

Unit Vector : A vector whose magnitude is one. In many applications of vectors it is helpful to find to find the unit vector with the same direction as a given vector. 1 3 The unit vector for w

5 4 3 Find the unit vector having the same direction as vector v. 5/5 4/5 3 3/5

Find the unit vector having the same direction as vector v. 4 -2

Find the unit vector having the same direction as vector v. 1 -3

P 751 39 – 45 odd, 47 – 50, 61-63,65,66

Airspeed Vector + Windspeed Vector = Groundspeed Vector Compass Heading True Bearingg Airspeed vector Airspeed – Speed relative to the air or speed if there was no wind. Compass Heading – The direction the plane would travel if there was no wind or the direction the engine is pushing the plane. This is the direction of the airspeed vector. Groundspeed – The speed of the plane relative to the ground or the speed of the plane with the wind. True Bearing or True Course – The direction the plane flies with the wind pushing it or the direction relative to the ground. This is the direction of the groundspeed vector. Wind Speed – Speed of the wind.

Directions and Bearings The direction to a point is stated as the number of degrees east or west of north or south. For example, the direction of A from O is N30ºE. B is N60ºW from O. C is S70ºE from O. D is S80ºW from O.                                                               Note: N30ºE means the direction is 30º east of north.

N E W S N E W S N E W S N E W S

Two forces F1 and F2, of magnitude 10 and 30 pounds respectively, act on an object. The direction of F1 is N20°E and the direction of F2 is N65°E. Find the magnitude and direction of the resultant force. 20° 10lbs 70° 20° 65° 30lbs 65° 25° 20° 65°

Two forces F1 and F2, of magnitude 10 and 30 pounds respectively, act on an object. The direction of F1 is N20°E and the direction of F2 is N65°E. Find the magnitude and direction of the resultant force. 20° 65° 37.74 22.08 37.74 lbs. 30.61

Two F1 and F2 forces, of magnitude 30 and 60 pounds respectively, act on an object. The direction of F1 is N10°E and the direction of F2 is N60°E. Find the magnitude and direction of the resultant force. 10° 30lbs 80° 10° 60° 60lbs 60° 30° 10° 60°

Two F1 and F2 forces, of magnitude 30 and 60 pounds respectively, act on an object. The direction of F1 is N10°E and the direction of F2 is N60°E. Find the magnitude and direction of the resultant force. 82.54 59.54 51.96 82.54 lbs.

A pilot whose plane can maintain an airspeed of 350 miles/hr sets the compass at N 15° E and is blown off course by a 90 mph wind blowing due east . What is the groundspeed and the true bearing of the plane. 15° 350 mph 75° 90 350 mph 15°

Angle with the positive x-axis 61.89° A pilot whose plane can maintain an airspeed of 350 miles/hr sets the compass at N 15° E and is blown off course by a 90 mph wind blowing due east . What is the groundspeed and the true bearing of the plane. 383.28 350 mph 338.07 15° 180.59 Groundspeed: 383.3 mph Angle with the positive x-axis 61.89° The groundspeed of the plane is 383.3 mph and the true bearing is N28°E.

A boat whose speed in still water is 20 mph is traveling across a river perpendicular to the 5 mph current. What is the actual speed the boat is traveling and at what angle is the boat blown off course. 5 mph 20 mph 20.62 The actual speed of the boat is 20.62 mph and it is blown off course at an angle of 14°

P 751 71 – 74, 83, 84

Wkst

An airplane pilot with an airspeed of 400 miles/hour orients her plane due Southwest (45 degrees South of West). The plane encounters a 100 mile/hour wind blowing towards the east. Determine the groundspeed and the true bearing of the plane. 225° 45° 400 mph 100 225° 45° 400 mph 100

Angle with positive x-axis = 237.12° An airplane pilot with an airspeed of 400 miles/hour orients her plane due Southwest (45 degrees South of West). The plane encounters a 100 mile/hour wind blowing towards the east. Determine the groundspeed and the true bearing of the plane. 225° 237.12° -182.84 45° 32.88° -282.84 400 mph 336.79 100 Groundspeed: 336.79 mph Angle with positive x-axis = 237.12° The groundspeed of the plane is 336.8 mph the true bearing of the plane is S33°W