Chapter 9-Vectors Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

Chapter 9-Vectors 9.1 Vectors in the Plane Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved INSERT FIGURE 9-1-1

Chapter 9-Vectors 9.1 Vectors in the Plane Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

Chapter 9-Vectors 9.1 Vectors in the Plane Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Vectors

Chapter 9-Vectors 9.1 Vectors in the Plane Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved DEFINTION: The sum v+w of two vectors v= and w= is formed by adding the vectors componentwise: Vector Algebra

Chapter 9-Vectors 9.1 Vectors in the Plane Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Add the vectors v = and w =. Vector Algebra DEFINITION: The zero vector 0 is the vector both of whose components are 0. DEFINITION: If v = is a vector and is a real number then we define the scalar multiplication of v by to be

Chapter 9-Vectors 9.1 Vectors in the Plane Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Length (or Magnitude) of a Vector THEOREM: If v is a vector and is a scalar, then

Chapter 9-Vectors 9.1 Vectors in the Plane Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Unit Vectors and Directions

Chapter 9-Vectors 9.1 Vectors in the Plane Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Unit Vectors and Directions DEFINITION: Let v and w be nonzero vectors. We say that v and w have the same direction if dir(v) =dir(w). We say that v and w are opposite in direction if dir(v) = −dir(w). We say that v and w are parallel if either (i) v and w have the same direction or (ii) v and w are opposite in direction. Although the zero vector 0 does not have a direction, it is conventional to say that 0 is parallel to every vector.

Chapter 9-Vectors 9.1 Vectors in the Plane Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Unit Vectors and Directions THEOREM: Vectors v and w are parallel if and only if at least one of the following two equations holds: (i) v = 0 or (ii) w = v for some scalar. Moreover, if v and w are both nonzero and w = v, then v and w have the same direction if 0 < and opposite directions if < 0.

Chapter 9-Vectors 9.1 Vectors in the Plane Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Unit Vectors and Directions EXAMPLE: For what value of a are the vectors and parallel?

Chapter 9-Vectors 9.1 Vectors in the Plane Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved An Application to Physics EXAMPLE: Two workers are each pulling on a rope attached to a dead tree stump. One pulls in the northerly direction with a force of 100 pounds and the other in the easterly direction with a force of 75 pounds. Compute the resultant force that is applied to the tree stump.

Chapter 9-Vectors 9.1 Vectors in the Plane Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Special Unit Vectors i and j i= and j= Suppose that v= and w=. Express v, w, and v+w in terms of i and j.

Chapter 9-Vectors 9.1 Vectors in the Plane Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Triangle Inequality EXAMPLE: Verify the Triangle Inequality for the vectors v = and w =.

Chapter 9-Vectors 9.1 Vectors in the Plane Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz

Chapter 9-Vectors 9.2 Vectors in Three- Dimensional Space Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Sketch the points (3,2,5), (2,3,-3), and (-1,-2,1).

Chapter 9-Vectors 9.2 Vectors in Three-Dimensional Space Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Distance THEOREM: The distance d(P,Q) between points P=(p 1, p 2, p 3 ) and Q=(q 1, q 2, q 3 ) is given by

Chapter 9-Vectors 9.2 Vectors in Three-Dimensional Space Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Distance EXAMPLE: Determine what set of points is described by the equation

Chapter 9-Vectors 9.2 Vectors in Three-Dimensional Space Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Distance DEFINITION: Let P 0 = (x 0, y 0, z 0 ) be a point in space and let r be a positive number. The set is the set of all points inside the sphere This set is called the open ball with center P 0 and radius r.

Chapter 9-Vectors 9.2 Vectors in Three-Dimensional Space Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Vectors in Space

Chapter 9-Vectors 9.2 Vectors in Three-Dimensional Space Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Vector Operations EXAMPLE: Suppose v= and w=. Calculate v+w, and sketch the three vectors. EXAMPLE: Suppose v=. Calculate 3v and -4v.

Chapter 9-Vectors 9.2 Vectors in Three-Dimensional Space Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Length of a Vector If v= is a vector, then the length or magnitude of v is defined to be

Chapter 9-Vectors 9.2 Vectors in Three-Dimensional Space Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Unit Vectors and Directions EXAMPLE: Is there a value of r for which u= is a unit vector? EXAMPLE: Suppose that v= and w=. Are there values of b and c for which v and w are parallel?

Chapter 9-Vectors 9.2 Vectors in Three-Dimensional Space Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz

Chapter 9-Vectors 9.3 The Dot Product and Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Algebraic Definition of the Dot Product DEFINITION: The dot product v  w of two vectors v and w is the sum of the products of corresponding entries of v and w. EXAMPLE: Let u =, v =, and w =. Calculate the dot products u · v, u · w, and v · w.

Chapter 9-Vectors 9.3 The Dot Product and Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Algebraic Definition of the Dot Product THEOREM: Suppose that u, v, and w are vectors and that is a scalar. The dot product satisfies the following elementary properties:

Chapter 9-Vectors 9.3 The Dot Product and Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Algebraic Definition of the Dot Product EXAMPLE: Let u= and v=. Calculate (u  v)u+(v  u)v

Chapter 9-Vectors 9.3 The Dot Product and Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved A Geometric Formula for the Dot Product THEOREM: Let v and w be nonzero vectors. Then the angle  between v and w satisfies the equation

Chapter 9-Vectors 9.3 The Dot Product and Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved A Geometric Formula for the Dot Product EXAMPLE: Calculate the angle between the two vectors v= and w=.

Chapter 9-Vectors 9.3 The Dot Product and Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved A Geometric Formula for the Dot Product Cauchy-Schwarz Inequality: EXAMPLE: Verify that the two vectors v = and w = satisfy the Cauchy-Schwarz Inequality.

Chapter 9-Vectors 9.3 The Dot Product and Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved A Geometric Formula for the Dot Product DEFINTION: Let  be the angle between nonzero vectors v and w. If  =  /2 then we say that the vectors v and w are orthogonal or mutually perpendicular.

Chapter 9-Vectors 9.3 The Dot Product and Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved A Geometric Formula for the Dot Product THEOREM: Let v and w be any vectors. Then: a) v and w are orthogonal if and only if v · w = 0. b) v and w are parallel if and only if c) If v and w are nonzero, and if  is the angle between them, then v and w are parallel if and only if  = 0 or  = . In this case, v and w have the same direction if  = 0 and opposite directions if  = .

Chapter 9-Vectors 9.3 The Dot Product and Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved A Geometric Formula for the Dot Product EXAMPLE: Consider the vectors u =, v =, and w =. Are any of these vectors orthogonal? Parallel?

Chapter 9-Vectors 9.3 The Dot Product and Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Projection THEOREM: If v and w are nonzero vectors then the projection P w (v) of v onto w is given by The length of P w (v) is given by

Chapter 9-Vectors 9.3 The Dot Product and Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Projection EXAMPLE: Let v = and w =. Calculate the projection of v onto w, the projection of w onto v, and calculate the lengths of these projections. Also calculate the component of v in the direction of w and the component of w in the direction of v.

Chapter 9-Vectors 9.3 The Dot Product and Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Projection and the Standard Basis Vectors EXAMPLE: Let v =. Calculate P i (v), P j (v), and P k (v) and express v as a linear combination of these projections.

Chapter 9-Vectors 9.3 The Dot Product and Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Direction Cosines and Direction Angles EXAMPLE: Calculate the direction cosines and direction angels for the vector

Chapter 9-Vectors 9.3 The Dot Product and Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Applications EXAMPLE: A tow truck pulls a disabled vehicle a total of 20,000 feet. In order to keep the vehicle in motion, the truck must apply a constant force of 3, 000 pounds. The hitch is set up so that the force is exerted at an angle of 30  with the horizontal. How much work is performed?

Chapter 9-Vectors 9.3 The Dot Product and Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. Calculate ·. 2. Use the arccosine to express the angle between and. 3. For what value of a are and perpendicular? 4. Calculate the projection of onto.

Chapter 9-Vectors 9.4 The Cross Product and Triple Product Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Cross Product of Two Spatial Vectors DEFINTION: If v = and w =, then we define their cross product v × w to be

Chapter 9-Vectors 9.4 The Cross Product and Triple Product Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Cross Product of Two Spatial Vectors THEOREM: If v and w are vectors, then v × w is orthogonal to both v and w. EXAMPLE: Let v = and w =. Calculate v × w. Verify that v × w is orthogonal to both v and w.

Chapter 9-Vectors 9.4 The Cross Product and Triple Product Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Relationship between Cross Products and Determinants If v = and w =, then EXAMPLE: Use a determinant to calculate the cross product of v = and w =.

Chapter 9-Vectors 9.4 The Cross Product and Triple Product Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Algebraic Properties of the Cross Product If u, v, and w are vectors and and  are scalars, then

Chapter 9-Vectors 9.4 The Cross Product and Triple Product Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Algebraic Properties of the Cross Product THEOREM: If v is any vector, then v×v = 0. More generally, if u and v are parallel vectors then u×v = 0. EXAMPLE: Give an example to show that the cross product does not satisfy a cancellation property. Give an example to show that the cross product does not satisfy the associative property.

Chapter 9-Vectors 9.4 The Cross Product and Triple Product Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved A Geometric Understanding of the Cross Product EXAMPLE: Find the standard unit normal for the pairs (i, j) and (j, k) and (k, i). Find also the standard unit normal for the pairs (j, i) and (k, j) and (i, k).

Chapter 9-Vectors 9.4 The Cross Product and Triple Product Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved A Geometric Understanding of the Cross Product THEOREM: Let v and w be vectors. Then a) b) If v and w are nonzero, then where   [0,  ] denotes the angle between v and w; c) v and w are parallel if and only if v × w = 0; d) If v and w are not parallel, then v × w points in the direction of the standard unit normal for the pair (v,w).

Chapter 9-Vectors 9.4 The Cross Product and Triple Product Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved A Geometric Understanding of the Cross Product EXAMPLE: Let v = and w =. What is the standard unit normal vector for (v,w)?

Chapter 9-Vectors 9.4 The Cross Product and Triple Product Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Cross Products and the Calculation of Area THEOREM: Suppose that v and w are nonparallel vectors. The area of the triangle determined by v and w is The area of the parallelogram determined by v and w is EXAMPLE: Find the area of the parallelogram determined by the vectors v= and w=.

Chapter 9-Vectors 9.4 The Cross Product and Triple Product Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Triple Scalar Product DEFINITION: If u, v, and w are given vectors, then we define their triple scalar product to be the number (u×v) · w. EXAMPLE: Calculate the triple scalar product of u =, v =, and w = in two different ways.

Chapter 9-Vectors 9.4 The Cross Product and Triple Product Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Triple Scalar Product THEOREM: The triple scalar product of u =, v =, and w = is given by the formula

Chapter 9-Vectors 9.4 The Cross Product and Triple Product Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Triple Scalar Product EXAMPLE: Use the determinant to calculate the volume of the parallelepiped determined by the vectors,, and.

Chapter 9-Vectors 9.4 The Cross Product and Triple Product Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Triple Scalar Product THEOREM: Three vectors u,v, and w are coplanar if and only if u · (v × w) = 0.

Chapter 9-Vectors 9.4 The Cross Product and Triple Product Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Triple Vector Products DEFINITION: If u, v, and w are given spatial vectors, then each of the vectors u × (v × w) and (u × v) × w is said to be a triple vector product of u, v, and w. EXAMPLE: Let v and w be perpendicular spatial vectors. Show that and

Chapter 9-Vectors 9.4 The Cross Product and Triple Product Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. Calculate ×. 2. Find the area of the parallelogram determined by and. 3. Find the standard unit normal vector for the ordered pair (, ). 4. True or false: a) v × w = w × v b) u × (v × w) = (u × v) × w c) u · v × w = u × v · w?

Chapter 9-Vectors 9.5 Lines and Planes in Space Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Cartesian Equations of Planes in Space THEOREM: Let V be a plane in space. Suppose that n= is a normal vector for V and that P 0 =(x 0,y 0,z 0 ) is a point on V. Then is a Cartesian equation for V.

Chapter 9-Vectors 9.5 Lines and Planes in Space Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Cartesian Equations of Planes in Space THEOREM: Suppose at least one of the coefficients A, B, C is nonzero. Then the solution set of the equation A(x−x 0 )+B(y−y 0 )+C(z−z 0 ) = 0 is the plane that has as a normal vector and passes through the point (x 0, y 0, z 0 ). The solution set of the equation Ax + By + Cz = D is a plane that has as a normal vector.

Chapter 9-Vectors 9.5 Lines and Planes in Space Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Cartesian Equations of Planes in Space EXAMPLE: Find an equation for the plane V passing through the points P = (2,−1, 4), Q = (3, 1, 2), and R = (6, 0, 5). EXAMPLE: Find the angle between the plane with Cartesian equation x − y − z = 7 and the plane with Cartesian equation −x + y − 3z = 6.

Chapter 9-Vectors 9.5 Lines and Planes in Space Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Parametric Equations of Planes in Space THEOREM: If P 0 = (x 0, y 0, z 0 ) is a point on a plane V and if u = and v = are any two nonparallel vectors that are perpendicular to a normal vector for V, then V consists precisely of those points (x, y, z) with coordinates that satisfy the vector equation When written coordinatewise, equation above yields parametric equations for V:

Chapter 9-Vectors 9.5 Lines and Planes in Space Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Parametric Equations of Planes in Space EXAMPLE: Find parametric equations for the plane V whose Cartesian equation is 3x − y + 2z = 7.

Chapter 9-Vectors 9.5 Lines and Planes in Space Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Parametric Equations of Lines in Space THEOREM: The line in space that passes through the point P 0 = (x 0, y 0, z 0 ) and is parallel to the vector m = has equation Here P = (x, y, z) is a variable point on the line. In coordinates the equation may be written as three parametric equations:

Chapter 9-Vectors 9.5 Lines and Planes in Space Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Parametric Equations of Lines in Space

Chapter 9-Vectors 9.5 Lines and Planes in Space Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Cartesian Equations of Line in Space

Chapter 9-Vectors 9.5 Lines and Planes in Space Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Cartesian Equations of Line in Space EXAMPLE: Find parametric equations of the line of intersection of the two planes x − 2y + z = 4 and 2x + y − z = 3.

Chapter 9-Vectors 9.5 Lines and Planes in Space Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Cartesian Equations of Line in Space EXAMPLE: Find parametric equations of the line of intersection of the two planes x − 2y + z = 4 and 2x + y − z = 3.

Chapter 9-Vectors 9.5 Lines and Planes in Space Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Calculating Distance THEOREM: Suppose that P = (x 0, y 0, z 0 ) is a point and that V is a plane. Let n = be a normal vector for V and let Q = (x 1, y 1, z 1 ) be any point on V. The distance between P and V is equal to

Chapter 9-Vectors 9.5 Lines and Planes in Space Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Calculating Distance EXAMPLE: Find the distance between the point P = (3,−8, 3) and the plane V whose Cartesian equation is 2x + y − 2z = 10.

Chapter 9-Vectors 9.5 Lines and Planes in Space Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz