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Chapter 9. Vectors and Oblique Triangles. 9.1 An Introduction to Vectors. A vector quantity is one that has direction as well as magnitude . For example, velocity describes the direction of the motion as well as the magnitude (the speed).
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Chapter 9 Vectors and Oblique Triangles
9.1 An Introduction to Vectors A vector quantity is one that has direction as well as magnitude. • For example, velocity describes the direction of the motion as well as the magnitude (the speed). A scalar quantity is one that has magnitude but no direction. • Some examples of scalar quantities are speed, time, area, mass.
A 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 Built-in feature in Equation Editor
Geometrically, vectors are like directed line segments. Each vector has an initial point and and a terminal point. • Q Terminal Point • Initial PointP Sometimes, vectors are expressed using the initial and terminal points.
C A B Two vectors are equal if they have the same _____________ and the same _______________________.
Addition of Vectors (Two Methods) The sum of any number of vectors is called the ____________________________. Two common ways of adding vectors graphically are the • POLYGON METHOD, and • PARALLELOGRAM METHOD.
B A Polygon Method To add vectors using the polygon method, position vectors so that they are tail to head. 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
B A 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
B A Polygon Method (cont.) This method can be used to add three or more vectors. Example: Add A + B + C C
Parallelogram Method To add two vectors using the parallelogram method, position vectors so that they are tail to tail, 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. B A
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.) B A Example: Add 2A + B
B A 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
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).
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.
y V=13.8 V Vy 63.5° x 0 Vx 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.
y V=13.8 V Vy 63.5° x 0 Vx To find the x- and y-components of V, we will use right triangle trigonometry. y-component x-component
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
y x 0 Example:Find the x- and y-components of the given vector by use of the trig functions. 1) 9750 N, = 243.0°
Example:Find the x- and y-components of the given vector by use of the trig functions. 2) 16.4 cm/s2, = 156.5° y x
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.
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.
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.
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.
To determine the measure of angle , you need to know the quadrant in which R lies. • If R lies in
Example Find the result of three vectors A, B, and C, such that
9.5 - 9.6 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
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.
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)
Case 1: Two angles and one side (ASA or SAA)
Case 2: Two sides and the angle opposite one of them (SSA)
Case 3: Two sides and the included angle (SAS)
Case 4: Three sides (SSS)
C b a B A c The Law of Sines Three equations for the price of one!
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.
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.
No TriangleIf , then side is not sufficiently long enough to form a triangle.
One Right TriangleIf , thenside is just long enough to form a right triangle.
Two TrianglesIf and , two distinct triangles can be formed from the given information.
