Countdown Begins….to PIE DAY!!!! Take-Out: HW #1 and protractor/ ruler HW #2: p 502 #3-10 Updates:

8.3: Vectors in Three-Dimensional Space Agenda Review HW #1 Last Minute Questions 8.1-8.2 Quiz 8.3: Vectors in Three-Dimensional Space Reflection

Review HW

Vectors If you are given the initial and terminal points of a vector, to find the ordered pair: If you are given the initial and terminal points of a vector, to find the magnitude: Use Pythagorean Theorem if you already have the vector in component form. ( Which is what you did on your Simon Says worksheet) Use the distance formula if you have the initial and terminal point

Whiteboards! Write the ordered pair that represents the vector from C (7, -3) to D ( -2, -1). Then find the magnitude of CD. <-9, 2>; sqrt ( 85)

Whiteboard! This question tests your vocabulary: Write a vector with the same magnitude as <3, 4> , but with a different direction. Write a vector with the same direction as <3, 4> , but with a different magnitude. . (5, 0) ( 6, 8)

Last Minute Questions

8.1-8.2 Quiz Raise your hand if you need a protractor or ruler? I will hand these out to you BUT I expect them to be returned to me once you are done with the quiz.

Learning Objectives By the end of this period you will be able to: Add and subtract vectors in three-dimensional space ( 2) Find the magnitude of vectors in 3D space

Take out a Paper.. Put U8L3 on the upper right hand corner. Title it: 8.3: Vectors in 3D Space

Vectors in 3D Space Vectors in three-dimensional space be described by coordinates in a way similar to the way we describe vectors in a plane. To show three real number lines intersecting at the zero point a figure with the x-axis appears to come out of the page to show depth.

Vectors in 3D Space

Vectors in 3D Space

Vectors in 3D Space http://hotmath.com/learning_activities/interactivities/3dplotter.swf

Plotting Points in 3D Space Example 1: Plot ( 3, 2, 4 ) Determine ( x, y, z ) First plot x point and y point. Draw dotted lines parallel to the x and y axis to connect the two points drawn.

Plotting Points in 3D Space Plot z Draw dotted lines parallel to the other axes

Plotting Points in 3D Space From those new points, draw lines parallel to the other axes. Where all the points meet is where the point is located.

Vectors in 3D Ordered triples can be represented as vectors. Example 2: a) Write the ordered triple that represents the vector X( 5, -3, 2) to Y (4, -5, 6). Think: How much is it moving on your x-axis, y-axis and z-axis?

Whiteboards! Plot ( -4, 3 , 2 )

Vectors in 3D Example 2: X( 5, -3, 2) to Y (4, -5, 6). Find the magnitude of XY Think: How did we find the magnitude when we had a vector with just ( x ,y ) We used the distance formula or Pythagorean Theorem! You can do distance OR just square each value

Whiteboard! Write the ordered triple that represents the vector T( -2, 4, 7) to M (-3, 5, 2). Then find the magnitude of TM.

Example 3 Find the ordered pair that represents 3p -2q if p=< 3, 0 , 4> and q = < 2 , 1, -1>

Whiteboard! If v = < 4, -3 , 5 > and w= < 2, 6, -1 > and z= < 3 , 0 ,4 > Find the following:

Whiteboard! What are the unit vectors?? Write what you think the unit vectors for j and i would be if we are talking about space ( 3D) Now for the z value we use k < 0, 0, 1> as our unit vector.

Example 4 Write AB as the sum of units for for A ( 5, 10, -3 ) and B ( -1, 4, -2).

Whiteboard Write AB as the sum of units for for A ( 5, 10, -3 ) and B ( -1, 4, -2).

Example 1(b) In the 12th Bristol International Kite Festival in England, Peter Lynn set a record for flying the worlds biggest kite, which had a lifting surface area of 630 square meters. Suppose the wind is blowing against the kite with a force of 100 N at an angle of 20° above the horizontal. a) Draw a diagram b) How much force is lifting the kite? Do this with your table-group

Example 1(c) After a rodeo, the owners of the bulls have to put them back into the paddocks. Suppose one owner is exerting force of 270 N due north and the other is pulling with a force of 360 N due east.   Draw a labeled diagram that represents the force Determine the resultant force exerted on the bull by the two owners. Find the angle the resultant force makes with the east- west axis. Do this with your table-group

Example 1(d) A large television screen inside a restaurant is supported equally on each side by two cables suspended from the ceiling at the restaurant. The cables form a 140° angle with each other. If the screen weighs 950 pounds, what is the force exerted by each of the cables on the screen? Hint: If vectors representing forces of equilibrium are drawn tip-to-tail, they will form a polygon.

Example 1(e) A boat is headed up a river at an angle of 25° with a rate of 20 feet per second. The current is heading down stream at a rate of 5 feet per second. The river is 100 feet wide. Express the velocity of the boat as a vector in component form without including the effect of the river. The actual motion of the boat is the combination of the motion of the boat and the current in the river. Find the vector that gives the actual motion of the boat. How long will it take the boat to cross the river? ( Hint: Use d= rt) How far up river will the boat arrive?

Example 1(f) With football season over, Mike and Dustin decided to go skiing. Unfortunately, the chair lift broke down when they were half way up the hill. If Mike and Dustin’s combined weight is 450 pounds, how many pounds of force are being applied to each of the cables? In the diagram, Mike and Dustin’s weight is represented by vector w and the amount of force applied to each of the cables is represented by vectors u and v. a. Find the force vector for w in component form. b. Find the force vector v. c. Write a system of equations so that the vectors are at equilibrium (sum of the force vectors equal zero). d. Solve the system in part (c) to find the forces applied to the two wires.

Ex 1 ( g) You are flying in a single engine plane from New Orleans to Houston. From New Orleans, you head due west towards Houston, which is 315 miles away, but shortly after take off your plane’s navigational equipment goes out. You press on and, as you reach the cruising altitude of 14,000 feet, you encounter a headwind blowing towards the southeast at 64 mph. The plane’s cruising speed is 210 mph, but the wind will affect the plane’s speed and direction. Draw and label a vector diagram for this situation. In particular, find and label 
the angle between the plane's velocity vector and the wind velocity vector. Add a vector representing the resulting velocity vector for the plane. Find the plane’s true speed and direction.