VECTORS The study of vectors is closely related to the study of such physical properties as force, motion, velocity, and other related topics. Vectors allow us to model certain characteristics of these phenomena with numbers that tell us their magnitude and direction.

SCALAR QUATITIES: Measurements involving such things as time, area, volume, energy, and temperature are called scalar measurements because each can be described adequately using their magnitude alone (with the appropriate units). 27 ft3 adequately describes the volume of a cube with side 3 ft. 980 F adequately describes the temperature of a person. 27ft3 and 980 F are called scalars.

Some properties such as force, velocity, and displacement require both magnitude and direction to be described completely. These quantities are called vector quantities. Example: You are driving due north at 45 miles per hour. The magnitude is the speed, 45 miles per hour. The direction of motion is due north.

NOTATION AND GEOMETRY OF VECTORS Two airplanes travel at 400 mph on a parallel course and in the same direction. This situation can be modeled using directed line segments. B The directed segments are drawn parallel with arrowheads pointing the same way to indicate direction of flight, while making them the same length indicates that the velocities are the same. The length of the vector models the magnitude of the velocity, while the arrowhead indicates the direction of travel. D A A Vectors are named using the initial and terminal points that define them as in Or with a bold, small case letter such as u or v. We may also write them as . C

The magnitude of the directed line segment is its length. Q P The magnitude of the directed line segment is its length. We indicate this by . is the distance from point P to point Q.

Reference Angle θr : For any angle θ in standard position, the acute angle θr formed by the terminal side and the x-axis is called the reference angle for θ. Find the reference angle, θr , for each of the following angles. a) θ = 3450 b) θ = -1350 c) 5800 a) θr = 150 b) θr = 450 c) θr = 400

Example Vectors that are equal have the same magnitude and direction. Use the distance formula to show that have the same magnitude.

One way to show that have the same direction is to find the slopes of the lines on which they lie. Verify to show that each vector has a slope of 3/2.

POSITION VECTORS For a vector v with initial point (x1, y1) and terminal point (x2, y2), the position vector for v is an equivalent vector with initial point (0,0) and terminal point (x2 – x1, y2 – y1). The vector in component form is denoted as , where a is the horizontal component and b is the vertical component.

Find the position vector for vector u and graph it. (3, 7) The position vector for u is: (-2, 3) 3 7 u The position vector v has (0,0) as its initial point, and (3, 7) as its terminal point. (-5, -4)

The magnitude of vector v = For a position vector v = ‹a, b› shown below at left and angle θr, observe the following: The magnitude of vector v = vertical component: horizontal component: a b x y θr

Finding the Magnitude and Direction Angle of a Vector Find their magnitudes. Graph each vector and name the quadrant where located. c. Find the angle θ for each vector (round to tenths of a degree).

is located in QIII. Why? 300

Find the Horizontal and Vertical Components of a Vector. x y Note: θr = 25⁰, therefore θ = 1800 + 250 = 2050. -19 250 -8.9

Vector Addition: Addition of vectors using the “tail – to – tip” method. Shift one vector (without changing its direction) so that its tail (initial point) is at the tip (terminal point) of the other vector. Given vectors u and v Tail of v to tip of u Tail of u to tip of v x y v u x y y x u v u + v u u + v u v v

Vector Addition: Add vectors v and w. Terminal point of w Initial point of v

APPLICATION OF VECTORS: A common example of a vector quantity is force. Other vector quantities that appear in engineering mechanics are moment, displacement, velocity, and acceleration. FORCE: The effect of one physical body on another physical body. The force effect between two bodies can be interpreted as a ‘push’ or ‘pull’ of one of the bodies on the other body.

RESULTANT FORCE: When two or more forces are added to obtain a single force, it produces the same effect as the original system of forces. This single vector is called the sum, or the resultant force of the original system of forces. The resultant force, FR , is the sum of F1 and F2 . y FR F1 F2 x

COMPONENTS OF A FORCE: Two or more forces acting on a particle may be replaced by a single force which has the same effect on the particle. Conversely, a single force F acting on a particle may be replaced by two or more forces which, together, have the same effect on the particle. These forces are called the components of the original force F.

RECTANGULAR COMPONENTS OF A FORCE. Often it is desirable to resolve a force into two components which are perpendicular to each other. In the figure below, the force F has been resolved into a component Fx and a component Fy. Fx and Fy are called rectangular components. y Fx = Fcosθ Fy = Fsinθ Fy F θ x Fx

We write F in the form F = -(655N)i + (459 N) j Example: A force of 800 N is exerted on a bolt A as shown. Determine the horizontal and vertical components of the force. Note: θR = 350; What is θ ? 350 F = 800 N A A Fy Fx F = 800 N 350 θ = 1450 Fx = Fcos1450 = 800cos1450 = -655 N Fy = Fsin1450 = 800sin145˚ = 459 N We write F in the form F = -(655N)i + (459 N) j

Vectors in the Rectangular Coordinate System

The i and j Unit Vectors. Vector i is the unit vector (vector of length 1) whose direction is along the positive x-axis. Vector j is the unit vector whose direction is along the positive y-axis. y Vectors in the rectangular coordinate system can be represented in terms of i and j. 1 j x i 1

b a x P = (a, b) v = ai + bj

A unit vector is defined to be a vector whose magnitude is one A unit vector is defined to be a vector whose magnitude is one. In many applications, it is useful to find the unit vector that has the same direction as a given vector.

Example: Find the unit vector in the same direction as v = 5i – 12j Example: Find the unit vector in the same direction as v = 5i – 12j. Then verify that the vector has magnitude 1.

Sketch the vector and find its magnitude. v = -3i + 4j a = -3 and b = 4. (-3, 4) v = 5

Example

Example

Operations with Vectors in Terms of i and j

Vector Subtraction: The difference of two vectors, u – v is defined as u – v = u + (-v): The terminal point of u coincides with the initial point of -v x y v u

Example

Unit Vectors

Example

Writing a Vector in Terms of Its Magnitude and Direction

Example

Application

Example

(a) (b) (c) (d)

(a) (b) (c) (d)

Converting from Rectangular Coordinates to Polar Coordinates. Radians. b) Polar Coordinates. A) RADIAN MEASURE We measure angles by determining the amount of rotation from the initial side to the terminal side. Two units of measurement for angles are degrees and radians.

An angle whose vertex is at the center of a circle Radian Measure An angle whose vertex is at the center of a circle is called a central angle. A central angle intercepts the arc of the circle from the initial side to the terminal side. A positive central angle that intercepts an arc of the circle of length equal to the radius of the circle has a measure of 1 radian. How many radians are there in a circle? r ө 6.28 How many degrees are there in 1 radian?

RELATIONSHIP BETWEEN DEGREES AND RADIANS: Degrees to radians: Radians to degrees:

Convert each angle from degrees to radians. a) 30˚ b) 90˚ c) -225˚ d) 55˚

Convert each angle in radians to degrees.

The foundation of the polar coordinate system is a horizontal ray that extends to the right. This ray is called the polar axis. The endpoint of the polar axis is called the pole. A point P in the polar coordinate system is designated by an ordered pair of numbers (r, θ). r is the directed distance form the pole to point P ( positive, negative, or zero). r pole polar axis θ P = (r, θ) θ is angle from the pole to P (in degrees or radians).

Plotting Points in Polar Coordinates.

Plot each point (r, θ) a) A(3, 450) C A b) B(-5, 1350) c) C(-3, -π/6) B

CONVERTING BETWEEN POLAR AND RECTANGULAR FORMS CONVERTING FROM POLAR TO RECTANGULAR COORDINATES. To convert the polar coordinates (r, θ) of a point to rectangular coordinates (x, y), use the equations x = rcosθ and y = rsinθ

Convert the polar coordinates of each point to its rectangular coordinates. a) (2, -30⁰ ) b) (-4, π/3) a) x = rcos(-30⁰) b) x= -4cos(π/3) = -4(1/2) = -2 y= -4 sin(π/3) =

CONVERTING FROM RECTANGULAR TO POLAR COORDINATES: To convert the rectangular coordinates (x, y) of a point to polar coordinates: Find the quadrant in which the given point (x, y) lies. 2) Use r =

Find the polar coordinates (r, θ) of the point P with r > 0 and 0 ≤ θ ≤ 2π, whose rectangular coordinates are (x, y) = The point is in quadrant 2. tanθ = The required polar coordinates are (2, 2π/3)

10.1 Three-Dimensional Coordinate Systems Distance Formula in Three Dimensions Equation of a sphere An equation of a sphere with center C(h, k, l) and radius r is (x – h)2 + (y – k)2 + (z – l)2

Copyright © 2011 Pearson Education, Inc Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Copyright © 2011 Pearson Education, Inc Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Copyright © 2011 Pearson Education, Inc Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Copyright © 2011 Pearson Education, Inc Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Copyright © 2011 Pearson Education, Inc Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Copyright © 2011 Pearson Education, Inc Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

R(0, 0, 12) Q(4, 0, 12) P(4, 9, 12)