Chapter 13: Vectors and the Geometry of Space 13.1 3-Dimensional Coordinate System 13.2 Vectors 13.3 The Dot Product 13.4 The Cross Product 13.5 Equations of Lines and Planes 13.6 Cylinders and Quadratic Surfaces I. 3-Dimensional Coordinate System 13.1 2-Dimensional Review Rectangular (Cartesian) Coordinates Quadrant II Quadrant III y-axis Quadrant I Quadrant IV x-axis x-axis where y = Set of points with coordinates (x, ) y-axis where x = Set of points with coordinates (, y) Origin is 1
All points are referred by two components (x,y) Ex: Plot (-3, 5) y x 3-Dimensions z-axis (x,y,z) Coordinate Axes Points on the x-axis: (,, ) Points on the y-axis: (,, ) Points on the z-axis: (,, ) y-axis Coordinate Planes xy-plane: x-axis yz-plane: "Right-Hand Rule" Origin: xz-plane: 2
The three coordinate planes divide space into eight octants z The first octant is where y Ex: Plot (3, 2, 5) x Ex: Describe the following. 1) z = 5 4) 2) x = -2 5) 3) y = 3 6) 3
Distance between P 1 (x 1, y 1, z 1 ) & P 2 (x 2, y 2, z 2 ) z P 2 (x 2, y 2, z 2 ) D d P 1 (x 1, y 1, z 1 ) y x Ex: Find the distance between (1, 2, 3) & (7, -3, 1). 4
Expanding distance formula into the standard equation of a sphere: Given a sphere with center (h, k, ) and radius r, the standard equation is given by Ex: Find the center and radius of the following sphere. 5
Ex: Describe the following 1) 4) 2) 5) 3) Ex: Use equations or inequalities to describe the following. 1) Circle of radius 4 with centered (0,0,0) that lies in the xz-plane. 2) Circle of radius 2 with center (1,2,3) that lies in the plane z = 3. 3) The cube in the first octant bounded by x = 3, y = 3, & z = 3. 6
II. Introduction to vectors 13.2 Vectors are used to describe applications that have magnitude and direction. Velocity vectors of air flowing over an airplane wing Force Vector Vector Notation Vectors can be represented by a directed line segment B (terminal point) v A (initial point) Vector denoted as v (boldface),v, or AB. Length or magnitude is denoted as AB, v, or v (not boldface). Two vectors are equivalent (or equal) if 7
Component form of vectors Ex: Given the vector in 1) Draw an equivalent that starts at the indicated pt in quadrant II. 2) Find the vector in component form. 3) Find the length an direction of the vector.. In 2-D y In 3-D z x y x Def: Zero Vector Q: "what about direction" 8
Ex: Find the vector with initial point (-2, 5) & terminal point (5, 1). "Use the graph paper" In 2-D In 3-D: Given initial point terminal point has and Ex: Find vector 1) from (3,-7) to (8,2) 2) from (4,0-2) to (1,6,-8) 9
Ex: Given that a vector starts at the origin, has length 5, and makes an angle of with the x-axis, find the component form. Use exact values. Operation on Vectors "Algebraic Definition" Given Given & & 10
"Parallelogram Law" Ex: Given &, Find and sketch. Ex: Given Find & Sketch 11
Ex: Given &, Find and sketch. III. Properties of Vectors a, b, c are vectors and c & d are scalars, then 1) a + b = b + a "Show examples" 2) a + (b + c) = (a + b) + c 3) a + 0 = 4) a + (-a) = 5) c(a + b) = 6) (c + d)a = 7) (cd)a = c(da) 8) 1a = a 12
Ex: Prove Property 1 (Assume a & b are in ) Ex: Prove Property 5 (Assume a & b are in ) 13
IV. Unit Vectors & Standard Base Vectors Def: Standard Base Vectors z y y x x Q: What is the length of the standard vectors? Writing a vector as a linear combination of the standard vectors: z y Ex: Write vectors as a linear combination of standard vectors. 1) x 2) 14
Ex: v = 1) Find a unit vector that is in the same direction as v. 2) Find a vector of length 10 that is in the same direction as v. Application Ex: You are pulling the handle of a wagon (see picture below) with a force F where F = 50 N. Find F in component form. Write F as a linear combination of the standard vecors. 15
V. Dot Product 13.3 So far we have discussed vector addition, subtraction & scalar multiplication. What about multication of vectors? Def: Dot (Inner, Scalar) Product Given Then & Ex: Find dot product. Given Then 1) 2) (3i + j) (5i +2j) & Note: The dot product of two vectors gives us a. Properties of dot product 1) Ex: Prove property 1 (Assume ) 2) 3) 4) 5) 16
Angle between vectors & alternative form of dot product. o z A? a B b y is angle between vectors a & b. Recall: Law of cosines B a c C b A x Ex: Prove Theorem:, where is angle between Corollary: Ex: Find angle between i - 2j - 2k & 6i + 3j + 2k. 17
Q: What can we say about if are orthogonal (perpendicular)? Hence, iff are orthogonal. Ex: Which pair of vectors is orthogonal? Ex: Find t so the the following vectors are perpendicular. 18
Direction angles & direction cosines In 2-D, we can define the direction of a vector with a single angle. y In 3-D, we need 3 vectors to define direction. z x y x Projection Idea: A wagon is being pulled such that it moves horizontally. There is a certain amount of the applied force that goes in the direction the wagon moves, which we call the scalar force. Direction Force is being applied. Direction wagon moves 19
Projection of b onto a: Previously we found Also, = scalar projection of b onto a. b a Therefore, Using what we found and the fact that the projection is a scalar multiple of a, Work If a constant force along the displacement vector, then the work done is defined by Ex: A wagon is being pulled with a constant force of 60 N at an angle of with the horizontal, so that it moves 50 meters along the ground. Find the work done. 20
VI. Cross Product 13.4 Note: Will use the determinant of a find the cross product of two vectors. matrix to give us an easier way to help Ex: Find Def: Cross Product Given & 21
Ex:, 1) Find 2) Find the dot product of and. Theorem: is orthogonal to both Proof: 22
Facts you need to remember! Given vectors. Fact 1: To test if the vectors are orthogonal, Fact 2: To find a third vector that is orthogonal to both, Theorem: is the angle between. Q: If are parallel ( ), then All though this is true, you should be able to see more easily that two vectors are parallel. Example: Why are the following vectors parallel? 23
Geometric Interpretation of : h Area of a parallelogram = (base)(height) h = Ex: Given the points P(1, -1, 0), Q(2, 1, -1), & R(-1, 1, 2). 1) Find the area of the triangle with vertices P, Q, & R. 2) Find a vector that is perpendicular to the plane that contains P, Q, & R. 24
parallelepiped VII. Equation of a line 13.5 Q: What does a linear equation look like in 2-D? Q: What does a linear equation look like in 3-D? z z y y x x 25
Equation of a line z l Developing vector form of line is a vector that points to when initial point is at origin. is a vector that runs x What you need! a point on the line. A vector parallel to the line. y (x, y, z) is any point on line l. Conclusion: Given a line and is parallel to l which contains the point, then a vector form is, where Parametic Equations for the line: Starting with Symmetric Equations for the line: 26
Ex: A line goes through the points (2,1,5) & (1,-2,7). Find the representation of the line in 1) Vector Form. 2) Parametric Equations. 3) Symmetric Equations. Ex: Where does the line in the last exercise intersect the xz-plane? 27
Ex: Determine whether the following pairs of lines intersect, are parallel, or are skew. 1) line 1: x = 1 + 2t, y = 2 + t, z = 3-3t line 2: x = 1 + s, y = 2 - s, z = 5 + 2s 2) line 1: x = 1 + 2t, y = 2 + t, z = 3-3t line 2: x = -6 + 3s, y = 9-2s, z = -6 + 2s 3) line 1: x = 1 + 2t, y = 2 + t, z = 3-3t line 2: x = 3-4s, y = 1-2s, z = 5 + 6s VIII. Equation of a Plane Ex: Use intercepts to graph the plane. 1) x + 3y + 2z = 12 2) 4x - 5y - 2z = 20 28
Deriving the equation of a plane: What you need! A point in the plane. A normal vector to the plane. Say P = (x,y,z) is any point in the plane. Then what can you say about Therefore, derive equation of plane Conclusion: Given a plane with the following A point in the plane. A normal vector to the plane. Ex: Find the equation of the plane that goes through the points (1,2,3), (1,0,-1), and (2,3,1) Eqn of plane is Remark: This equation can be written in the standard form 29
Ex: Given the two lines, 1) Find the point the two lines intersect. 2) Find an equation of the plane that contains these lines. Ex: Find the equation of the plane that contains the given two lines. line 1: line 2: 30
Observation: Given the plane n = is a normal to the plane. Ex: Given the planes x - 3y + z = 5 & x + y - 2z = 4 1) Find the parametric equations that is the intersection of the planes. 2) Find the angle between the planes. Ex: Find the equation of the plane that contains the point (1, 2, 3) and is parallel to 2x - 3y + z = 10. 31
Ex: Find the equation of the plane that contains the point (4, 5, 1) and is perpendicular to the line x = 1 + 2t, y = 3 + 2t, z = -4t. IX. Cylinders 13.6 Recall: In, graph the following. 1) y = 3 (Q: Does x exist in this equation?) 3) 2) x = 5 (Q: Does y exist in this equation?) 32
Returning to 1) 2) 3) Def: A cylinder is a surface that consists of all lines (called rulings) that are parallel to a given line and pass through a given plane. Ex: Graph Note: If a 3-D equation is not showing one of the variables, the result will be a cylinder. 33
X. Quadratic Surfaces The existence of these terms cause rotations & with a substitution can be transformed into We will use traces to help us graph such surfaces. Def: Traces are the curves of intersection between the surface & planes parallel to the coordinate planes. Ex: Use traces to sketch the following. "Make sure to show sketches of some traces" Ellipsoid 34
Elliptic Paraboloid Elliptic Paraboloid 35
Hyperbolic Paraboloid Hyperboloid of One Sheet 36
Hyperboloid of Two Sheets 37
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