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**Vectors and Oblique Triangles**

Chapter 9 Vectors and Oblique Triangles

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**9.1 An Introduction to Vectors**

MAT 205 SP09 9.1 An Introduction to Vectors 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.

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MAT 205 SP09 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

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MAT 205 SP09 Geometrically, vectors are like directed line segments. Each vector has an initial point and and a terminal point. • Q Terminal Point • Initial Point P Sometimes, vectors are expressed using the initial and terminal points.

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**Two vectors are equal if they have the same _____________ and the same _______________________.**

B We write:

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**Addition of Vectors (Two Methods)**

MAT 205 SP09 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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MAT 205 SP09 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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MAT 205 SP09 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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MAT 205 SP09 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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MAT 205 SP09 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**

MAT 205 SP09 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.) Example: Add 2A + B A B

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**Subtraction of Vectors**

MAT 205 SP09 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. Example: Find 2A - B A 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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**9.2 – 9.3 Components of Vectors**

MAT 205 SP09 9.2 – 9.3 Components of Vectors 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.

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MAT 205 SP09 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. x y V Vx Vy 63.5° V=13.8 Vector V, of magnitude 13.8 and direction 63.5°, and its components directed along the axes.

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MAT 205 SP09 To find the x- and y-components of V, we will use right triangle trigonometry. y-component x-component x y V Vx Vy 63.5° V=13.8

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

MAT 205 SP09 To Resolve a Vector Into its x- and y-components: Place vector V with initial point at origin such that its direction is given by an angle in standard position. Calculate the x-component by Vx = V cos Calculate the y-component by Vy = V sin

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MAT 205 SP09 Example: Find the x- and y-components of the given vector by use of the trig functions. 1) N, = 243.0° x y

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MAT 205 SP09 Example: Find the x- and y-components of the given vector by use of the trig functions. 2) cm/s2, = 156.5° y x

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MAT 205 SP09 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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MAT 205 SP09 From the text: P. 262 # 28

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**Vector Addition by Components**

MAT 205 SP09 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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**To determine the measure of angle , you need to know the quadrant in which R lies.**

If R lies in

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

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MAT 205 SP09 From the text: p. 267 # 8, 28

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MAT 205 SP09

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

MAT 205 SP09 The Law of Sines and The Law of Cosines 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

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**Every triangle has 3 sides and 3 angles. **

MAT 205 SP09 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: **

MAT 205 SP09 Here are the four possible combinations of parts: Two angles and one side (ASA or SAA) Two sides and the angle opposite one of them (SSA) Two sides and the included angle (SAS) Three sides (SSS)

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**Two angles and one side (ASA or SAA)**

MAT 205 SP09 Case 1: Two angles and one side (ASA or SAA)

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

MAT 205 SP09 Case 2: Two sides and the angle opposite one of them (SSA)

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**Two sides and the included angle (SAS)**

MAT 205 SP09 Case 3: Two sides and the included angle (SAS)

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MAT 205 SP09 Case 4: Three sides (SSS)

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**The Law of Sines Three equations for the price of one! MAT 205 SP09 C**

B A C c b a The Law of Sines Three equations for the price of one!

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**Solving Case 1: ASA or SAA**

Solve the triangle:

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**Solving Case 1: ASA or SAA**

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**Example using Law of Sines**

MAT 205 SP09 Example using Law of Sines 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.

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Continued from above

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MAT 205 SP09 Solving Case 2: SSA 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.

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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.**

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**One Right Triangle If , then side is just long enough to form a right triangle.**

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

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**One Triangle If , only one triangle can be formed.**

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Continued from above

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Continued from above

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MAT 205 SP09

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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 b a 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: 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: Use the Law of Cosines to find the third side. 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.)} Find the third angle by subtracting the two known angles from 180.

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Solving Case 3: SAS Example: Solve ABC where a = 184, b = 125, and C = 27.2.

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Continued from above

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Solving Case 3: SAS Example: Solve ABC where b = 16.4, c = 10.6, and A = 128.5.

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Continued from above

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**Guidelines for Solving Case 4: SSS**

When given three sides, follow these steps: Use the Law of Cosines to find the LARGEST ANGLE *(opposite the largest side). Use the Law of Sines to find either of the two remaining angles. Find the third angle by subtracting the two known angles from 180.

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Solving Case 4: SSS Example: Solve ABC where a = 128, b = 146, and c = 222.

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Continued from above

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**(Let bold red represent the given info)**

When to use what…… (Let bold red represent the given info) SAS AAS ASA Be careful!! May have 0, 1, or 2 solutions. SSS SSA Use Law of Sines Use Law of Cosines

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**To nearest tenth of a mile.**

To nearest tenth of a degree. To nearest minute.

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MAT 205 SP09 Continued from above

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