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Chapter 9

Vectors and Oblique Triangles

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A vector quantity is one that has ____________ as well as _________________.

For example, velocity describes the direction of the motion as well as the magnitude (the speed).

A scalar quantity is one that has ___________ but no ______________.

Some examples of scalar quantities are speed, time, area, mass.

9.1 An Introduction to Vectors

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Representing Vectors

In most textbooks, vectors are written in boldface capital letters. The scalar magnitude is written in lightface italic type.

So, B is understood to represent a vector quantity, having magnitude and direction, while B is understood to be a scalar quantity, having magnitude but no direction.

When handwriting a vector, place an arrow over the letter to represent a vector.

Write

A A

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Geometrically, vectors are like directed line segments.

Each vector has an initial point and and a terminal point.

•

•

Initial Point P

Q Terminal Point

Sometimes, vectors are expressed using the initial and terminal points.

PQ��������������

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Two vectors are equal if they have the same _____________

and the same _______________________.

A

C

B

We write:

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Addition of Vectors (Two Methods)

The sum of any number of vectors is called the

____________________________, usually represented

as ______.

Two common ways of adding vectors graphically are the POLYGON METHOD, and

PARALLELOGRAM METHOD.

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Polygon Method

To add vectors using the polygon method, position vectors so that they are tail (dot) to head (arrow).

The resultant is the vector from the initial point (tail) of the first vector to the terminal point (head) of the second.

When you move the vector(s), make sure that the magnitude and direction remain unchanged!

We use graph paper or a ruler and protractor to do this.

Example: Add A + B

A

B

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Polygon Method (cont.)

Vector addition is ________________________, which means that the order in which you add the vectors will not affect the sum.

Example: Add B + A

A

B

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Polygon Method (cont.)

This method can be used to add three or more vectors.

Example: Add A + B + C

A

B

C

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Parallelogram Method

To add two vectors using the parallelogram method, position vectors so that they are tail to tail (dot to dot), by letting the two vectors form the sides of a parallelogram.

The resultant is the diagonal of the parallelogram. The initial point of the resultant is the same as the initial points of each of the vectors being added.

A

B

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Scalar Multiplication

If n is a scalar number (no direction) and A is a vector, then nA is a vector that is in the same direction as A but whose magnitude is n times greater than A. (Graphically, we draw this vector n times longer than A.)

A

B

Example: Add 2A + B

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Subtraction of Vectors

Subtraction of vectors is accomplished by adding the opposite. A B = A + (B)

where –B is the vector with same magnitude as B but opposite direction.

A

B

Example: Find 2A - B

Label vectors appropriately!

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Force, velocity, and displacement are three very important vector quantities.

Force is expressed with magnitude (in Newtons) and direction (the angle at which it acts upon an object).

Velocity is expressed with magnitude (speed) and direction (angle or compass direction).

Displacement is expressed with magnitude (distance) and direction (angle or compass direction).

Do classwork: Representing Vectors Graphically

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Any vector can be replaced by two vectors which, acting together, duplicate the effect of the original vector. They are called components of the vector.

The components are usually chosen perpendicular to each other. These are called rectangular components.

The process of finding these components of a vector is called resolving the vector into its components.

9.2 – 9.3 Components of Vectors

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We will resolve a vector into its x- and y-components by placing the initial point of the vector at the origin of the rectangular coordinate plane and giving its direction by an angle in standard position.

Vector V, of magnitude 13.8 and direction 63.5°, and its components directed along the axes.

x

y

0

V

Vx

Vy

63.5°

V=13.8

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To find the x- and y-components of V, we will use right triangle trigonometry.

x

y

0

V

Vx

Vy

63.5°

V=13.8

cos 63.5 xV

V sin 63.5 yV

V

x-component y-component

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To Resolve a Vector Into its x- and y-components:

1. Place vector V with initial point at origin such that its direction is given by an angle in standard position.

2. Calculate the x-component by Vx = V cos

3. Calculate the y-component by Vy = V sin

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Example: Find the x- and y-components of the given vector by use of the trig functions.

1) 9750 N, = 243.0°

x

y

0

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Example: Find the x- and y-components of the given vector by use of the trig functions.

2) 16.4 cm/s2, = 156.5°

x

y

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A cable exerts a force of 558 N at an angle of 47.2° with the horizontal. Resolve this into its horizontal and vertical components.

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From the text: P. 262 # 28

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Vector Addition by Components

We can use this idea of vector components to find the resultant of any two perpendicular vectors.

Example:

If the components of vector A are Ax = 735 and Ay = 593, find the magnitude of A and the angle it makes with the x axis.

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Example:

Add perpendicular vectors A and B, given A = 4.85 and B =6.27

Find the magnitude and the angle that the resultant makes with vector A.

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Adding Non-Perpendicular Vectors

Place each vector with its tail at the origin

Resolve each vector into its x- and y-components

Add the x-components together to get Rx

Add the y-components together to get Ry

Use the Pythagorean theorem to find the magnitude of the resultant.

Use the inverse tangent function to help find the angle that gives the direction of the resultant.

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x yR R R

1tan yref

x

R

R

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To determine the measure of angle , you need to know the quadrant in which R lies.

If R lies in

Quadrant I 0& 0 :

Quadrant II 0& 0 : 180

Quadrant III 0& 0 : 180

Quadrant IV 0& 0 : 360

x y ref

x y ref

x y ref

x y ref

R R

R R

R R

R R

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ExampleFind the resultant of three vectors A, B, and C, such that

6.34, 29.5 ; 4.82, 47.2 ; 5.52, 73.0A B CA B and C

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From the text: p. 267 # 8, 28

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In this section, we will work with oblique triangles

triangles that do NOT contain a right angle.

An oblique triangle has either: three acute angles

two acute angles and one obtuse angle

or

9.5 - 9.6 The Law of Sines and The Law of Cosines

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Every triangle has 3 sides and 3 angles.

To solve a triangle means to find the lengths of its sides and the measures of its angles.

To do this, we need to know at least three of these parts, and at least one of them must be a side.

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Here are the four possible combinations of parts:

1. Two angles and one side (ASA or SAA)

2. Two sides and the angle opposite one of them (SSA)

3. Two sides and the included angle (SAS)

4. Three sides (SSS)

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Case 1: Two angles and one side (ASA or SAA)

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Case 2:

Two sides and the angle opposite one of them (SSA)

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Case 3:

Two sides and the included angle (SAS)

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Case 4:

Three sides (SSS)

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BA

C

c

ba

sin sin sin

a b c

A B C

Three equations for the price of one!

The Law of Sines

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Solving Case 1: ASA or SAA

Solve the triangle: 35.0 , 15.0 , 5.00A B c

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Solving Case 1: ASA or SAA

40.0 , 60.0 , 4.00A B a

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A ship takes a sighting on two buoys. At a certain instant,

the bearing of buoy A is N 44.23° W, and that of buoy B is N

62.17° E. The distance between the buoys is 3.60 km, and

the bearing of B from A is N 87.87° E. Find the distance of

the ship from each buoy.

Example using Law of Sines

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Continued from above

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In this case, we are given two sides and an angle opposite.

This is called the AMBIGUOUS CASE.

That is because it may yield no solution, one solution, or two solutions, depending on the given information.

Solving Case 2: SSA

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SSA --- The Ambiguous Case

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No Triangle If , then side is not sufficiently long enough to form a triangle.

sina h b A a

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One Right Triangle If , thenside is just long enough to form a right triangle.

sina h b A a

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Two Triangles If and , two distinct triangles can be formed from the given information.

sinh b A a a b

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One Triangle If , only one triangle can be formed.

a b

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3.0, 2.0, 40a b A

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Continued from above

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6.0, 8.0, 35a b A

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Continued from above

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1.0, 2.0, 50a b A

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Making fairly accurate sketches can help you to determine the number of solutions.

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Example: Solve ABC where A = 27.6, a =112, and c = 165.

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Continued from above

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To deal with Case 3 (SAS) and Case 4 (SSS), we do not have enough information to use the Law of Sines.

So, it is time to call in the Law of Cosines.

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B

A

C

c

ba

2 2 2

2 2 2

2 2 2

2 cos

2 cos

2 cos

a b c bc A

b a c ac B

c a b ab C

The Law of Cosines

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Using Law of Cosines to Find the Measure of an Angle

*To find the angle using Law of Cosines, you will need to solve the Law of Cosines formula for CosA, CosB, or CosC.

For example, if you want to find the measure of angle C, you would solve the following equation for CosC:

2 2 2 2 cosc a b ab C

2 2 22 cosab C a b c 2 2 2

cos2

a b cC

ab

To solve for C, you would take the cos-1 of both sides.

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Guidelines for Solving Case 3: SAS

When given two sides and the included angle, follow these steps:

1. Use the Law of Cosines to find the third side.

2. Use the Law of Cosines to find one of the remaining angles.

{You could use the Law of Sines here, but you must be careful due to the ambiguous situation. To keep out of trouble, find the SMALLER of the two remaining angles (It is the one opposite the shorter side.)}

3. Find the third angle by subtracting the two known angles from 180.

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Example: Solve ABC where a = 184, b = 125, and C = 27.2.

Solving Case 3: SAS

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Continued from above

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Example: Solve ABC where b = 16.4, c = 10.6, and A = 128.5.

Solving Case 3: SAS

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Continued from above

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Guidelines for Solving Case 4: SSS

When given three sides, follow these steps:

1. Use the Law of Cosines to find the LARGEST ANGLE *(opposite the largest side).

2. Use the Law of Sines to find either of the two remaining angles.

3. Find the third angle by subtracting the two known angles from 180.

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Example: Solve ABC where a = 128, b = 146, and c = 222.

Solving Case 4: SSS

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Continued from above

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When to use what……(Let bold red represent the given info)

Use Law of Sines

SSS

SSA

SAS

ASA

AAS

Use Law of Cosines

Be careful!! May have 0, 1, or 2 solutions.

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To nearest tenth of a mile.

To nearest tenth of a degree.

To nearest minute.

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Continued from above