Review of Trigonometry - MDHS Math Department...

Lesson1.notebook November 27, 2012

Unit 1 Representing and Operations with Vectors

Algebra

Over the years you have come to accept various mathematical concepts or properties:

Communative Property

Associative Property

Distributive Property

Identities

Inverse Operations

Problem Solving Skills 4 basic steps to effective problem solving:

1) Understand the problem:

2) Think of a plan:

3) Carry out the plan.

4) Reflect on solution:


Review of TrigonometryPythagorean Theorem

Right Angle Trigonometric Ratios

Sine Law

Cosine Law


CAST Rule

Example: From a cliff 50 m high, a person can see 2 boats. One boat is S50W at an angle of depression of 30 degrees. The second boat is S60E and is at an angle of depression of 45 degrees. How far apart are the boats?


Vector ConceptsToday's goal: I can explain the difference between a vector and scalar and I can describe the properties and notation associated with vectors.

Vectors

Scalar

Examples:

Vector Notation


Angles Between Vectors

Scalar Multiplication

Zero Vector

Unit Vectors

Examples: List all equal vectors in the given diagram.


Example: For the given number line, state equivalent vectors.

A 3 B 2 C 5 D 7 E

Homework: pg 121 #1 6, 8, 9Handoutpg 127 #1,2,4,6,8,9,13


Algebraic VectorsToday's goal: I can write a vector in both ordered pair and unit vector notation and complete the associated operations using algebraic methods.

There are two algebraic forms that we can use to represent vectors:

Ordered Pair Notation Unit Vector Notation

In 3 space:

The Direction Cosines for a vector in 3 space are:

From our grade 11 trigonometric identity:It now becomes:


Example: Determine the unit vector notation for the following vectors.A) u = [ 9, 12] B) v = [3, 5, 8]

Example: Write the following in order pair notation.A) |u| = 20, [N20W] B) |v| = 50, θ = 230


Example: Determine the magnitude, direction and unit vector for the following:

A) u = [2, 6] B) v = [1, 3, 9]

For 3 space we can break it into "areas" or "planes". For example:

HWpage 166 # 1 8, 11, 13, 14, 16 19, 23


Operations with Algebraic Vectors

Today's goal: I can extend my knowledge of algebraic vectors to the concepts of adding and subtracting, making the connection with currently understood algebra rules.

Two Algebraic vectors are equal if and only if their Cartesian components are equal.

Collinear vectors are any two vectors that lie on the same line.Which means:

Operations with Algebraic Vectors

Scalar multiplication: k(ai + bj) =

k[a, b] =

Vector addition/subtraction:ai + bj (ci dj) =

[a, b] [c, d] =

Example: If u = [3,8] and v = [2, 6], determine:

A) w if w = 3u + 2v B) w if w = 2u + 1/2v

C) |u + 3v|


Example: Given points P(6, 1), Q(2, 1) and R(3, 4), determine:A) QP B) |RP|

Example: Given points P(3, 1, 4), Q(3, 1, 5) and R(7, 4, 2), determine:A) QP B) |RP|


Example: Determine if the following points are collinear. Justify your answer.

P(2, 3, 6) Q(8, 1, 10) R(5, 2, 8)

HWpage 172 #2 10, 13, 14


Vector LawsToday's goal: I can apply the rules of vector addition to applications that model realworld situations, preparing me to apply them in a physics context.

If u and v are vectors:

1) Closure

2) Communative

3) Associative

4) Distributive (if a and b are scalars)

5) Identity

Triangle Law of Vector Addition

When adding/subtracting two vectors:

1) Draw the two vectors head to tail.

2) The "sum" or resultant is a vector that is measured from the tail of the first vector to the head of the second vector.


Parallelogram Law of Vector Addition

When adding/subtracting two vectors:

1) Draw the vectors tail to tail.

2) Create a parallelogram using the two vectors as two sides of the parallelogram.

3) Draw the resultant vector which extends from the tails to the heads of the vectors or it is the diagonal of the parallelogram.

Example: if |u| = 10 units, |v| = 5 units and the angle between the two vectors when placed tail to tail is 30 degrees, determine the resultant.


Geometric VectorsExample: Determine the resultant of 3u 2v + 5w.

Algebraic VectorsTo use algebraic vectors we need to review the right angle triangle and discuss components.


Example: if |u| = 10 units, |v| = 5 units and the angle between the two vectors when placed tail to tail is 30 degrees, determine the resultant.

Algebraic Vectors (Coordinate Form)Example: Determine the resultant of 3u 2v + 5w.


Determine u given the following vectors.

45 75

u = 25v = 10R = 15

Homework:

Pg 133, # 1, 3ab, 4 6, 10 12, 20Handout


Force and Velocity of a VectorToday's goal: I can apply previous understanding of vector addition and subtraction and apply them to realworld situations involving physics.

ForceFor this section everyone needs to "know" Newton's Three Laws:

1)

2)

3)

Example: A 5 kg sign is suspended as shown. Determine the tension in each wire.

45 60


45 60

Example: Chris and Josh are pulling a sled as shown. If Josh is pulling with 70 N of force, determine with what force Chris must exert so the sled is kept travelling straight.

3050

70 N

x


3050

70 N

x

Example: Determine the resulting force.


Velocity as a Vector

Velocity is a measure of speed (magnitude) and direction. We can use our knowledge of vectors to solve questions involving velocity.

Relative Velocity is the velocity an observer perceives other objects to be travelling at.

Complete the following chart:

Bus:Train:Car:

My Location Bus Train Car


Relative Motion Equation: vrel = vA + vB where,vrel

vA

vB

Example: A plane is travelling 400 km/h [N60E]. The pilot set a course of 500 km/h [N30E]. Determine the velocity of the wind.


Example: A ship steering 15 km/h [N70E]. If the current is 5 km/h [W10N], determine the relative velocity of the boat when standing on shore.

Homework:

pg 141, # 24, 610, 1315, 18, 24, 25pg 149, # 2 8, 13


The Dot Product of Two Vectors

Today's goal: I can recognize the need for the Dot Product and situations it is required. I can then apply it using previously discussed methods.

Example: Determine the amount of work to move the object from point A to point B in the diagram where:

f applied forceAB displacement vector

HINT: W = force x displacementWork is also a scalar!

A

Bf

20

θθ

General Notes on the Dot Product (also known as the Scalar Product)

For non zero vectors u and v, we define the Dot Product as:

u v = |u||v|cosθ, where θ is a value 0≤θ≤180.

If u or v is the zero vector, than u v = 0.

If θ=90, then cosθ = 0. If u and v are perpendicular to each other, the dot product is zero.


Properties of the Dot Product

1) For non zero vectors u and v, if u v = 0, the vectors are perpendicular.

2) For any vectors u and v, u v = v u. The Dot Product is communative.

3) For any vector u, u u = |u|2.

4) For any vectors u and v with a scalar value k, k εR,(ku) v = k(u v) = u (kv)

5) For any vectors u, v, and w,u (v + w) = u v + u w

Applying the Dot Product

If u = uxi + uyj + uzk and v = vxi + vyj + vzk then,

Multiplying unit vectors:

Performing the Dot Product:

u v =


Summary:

If u = [a, b, c] and v = [d, e, f] then:

u v =

Example: Determine the Dot Product.

1) u = [3,7] and v = [4, 2] 2) u = 2i + 3j 4k and v = i + 6j + 2k


Example: Determine the angle between the two vectors.

1) u = [2,4] and v = [7, 2] 2) u = [5, 2, 1] and v = [10, 3, 4]

Example: Determine the value of z to make the vectors perpendicular.

u = [2, 7, 6] and v = [5, 4, z]

Homework:

page 178 #1 15, 21


The Cross Product of Two Vectors

Today's goal: I can identify when the Cross Product would be required and how in principle it works. I can use the appropriate skills to determine its value.

Example: A wrench is tightening a nut. Discuss the forces involved.

Magnitude will depend on:

1)

2)

Resulting formula:

Why is torque a vector?

The Right Hand Rule

The Right Hand Rule is applied because we can model 3 vectors (ie. applied force, Torque or Moment, and lever length) using fingers from our hand.

In general, we point our fingers along vector u (ie. the lever length) and then curl our fingers in the direction v (ie. applied force) for angles of 0≤θ≤180. The direction of our thumb then indicates the direction of the torque or moment (ie. is the bolt or screw tightening or loosening from the board).


Notation

u x v = |u||v|sinθn, where:

General Properties

1) u x v = (v x u) Not communative

2) For nonzero vectors u and v, both vectors are collinear if, and only if, u x v = 0.

3) Two vectors are not collinear if u x v ≠ 0.

4) For vectors u, v, and w:u x (v + w) =(u + v) x w =ku x v =

Applying the Cross Product

For vectors u = [ux, uy, uz] and v = [vx, vy, vz]:

u x v =

=

Easier forms to help remember:


Example: Determine the Cross Product.

1) |u| = 5, |v| = 10, θ = 30 2) u = [3, 6, 2] and v = [2, 4, 8]

3) u = 2i 3j + 5k and v = 7i + 2j 3k

Example: Determine a vector perpendicular to u = [1, 3, 2] and v = [3, 2, 1]. Justify your answer.


Example: If |u x v| = 34, |u| = 4, and |v| = 9 the find θ.

Example: Find a x b where a = 2i 3j + 5k and b = 7i + 2j 3k.

Homework:page 185 #1 10, 14


Applications of the Dot and Cross Product

Today's goal: I can determine when the Dot and Cross Product need to be applied and understand how they are used in a variety of realworld applications that are modelled in a simple format.

The Dot and Cross product have a variety of applications where they can be applied individually or in combination.

Determining 3 Space

What is 3 space?

Example: Determine if the given vectors are the basis for space:

u = [2, 3, 5] v = [1, 0, 3] w = [4, 1, 0]


ProjectionsWe talked about the x and y components of a vector also being the projection (or shadow) onto the x and y axis. What about the projection of one vector onto another? For example...

Area of a ParallelogramWith drafting programs, they often have the ability to calculate the area of a given shape. This is how they work...

h

b

wArea =


Volume of a Parallelepiped (or any other rectangular prism)Similar to area, programs can also calculate volume. This is how they work...

Volume =

Work

Torque


Example: If u = [1, 3, 6] and v = [2, 1, 3], find the projection of u onto v and the magnitude of the projection.

Example: A parallelogram has vertices A(1, 1, 1), B(4, 3, 2), C(1, 2, 3) and D(2, 0, 4). Determine the area.


Example: Use the Dot Product to prove the cosine law.

Example: Determine the Torque for the following.


Homework:Proof questions:

1) Prove that: |u x v|2 = |u|2|v|2 (u v)2

2) Prove that: |a + b|2 + |a b|2 = 2|a|2 + 2|b|2

3) Prove that: |a|2|b|2 (a b)2 = 0

4) ABC is an isosceles triangle with base BC. Use vectors to prove that the median from vertex A is perpendicular to the base.

page 192 #1 3, 7, 9 18

Unit Test:

End date of unit:

Test dates:

Review dates:

Review material: Handoutpg 153 (pick and choose)pg 155 (pick and choose)pg 194 (pick and choose)pg 197 (pick and choose)

Review of Trigonometry - MDHS Math Department...

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