which has a check digit of 9. This is consistent with the first nine digits of the ISBN, since

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1 vector Then the check digit c is computed using the following procedure: 1. Form the dot product. 2. Divide by 11, thereby producing a remainder c that is an integer between 0 and 10, inclusive. The check digit is taken to be c, with the proviso that is written as X to avoid double digits. For example, the ISBN of the brief edition of Calculus, sixth edition, by Howard Anton is which has a check digit of 9. This is consistent with the first nine digits of the ISBN, since Dividing 152 by 11 produces a quotient of 13 and a remainder of 9, so the check digit is. If an electronic order is placed for a book with a certain ISBN, then the warehouse can use the above procedure to verify that the check digit is consistent with the first nine digits, thereby reducing the possibility of a costly shipping error. Concept Review Norm (or length or magnitude) of a vector Unit vector Normalized vector Standard unit vectors Distance between points in Angle between two vectors in Dot product (or Euclidean inner product) of two vectors in Cauchy-Schwarz inequality Triangle inequality Parallelogram equation for vectors Skills Compute the norm of a vector in. Determine whether a given vector in is a unit vector. Normalize a nonzero vector in. Determine the distance between two vectors in. Compute the dot product of two vectors in. Compute the angle between two nonzero vectors in. Prove basic properties pertaining to norms and dot products (Theorems and ).

2 Exercise Set 3.2 In Exercises 1 2, find the norm of v, a unit vector that has the same direction as v, and a unit vector that is oppositely directed to v In Exercises 3 4, evaluate the given expression with. and 3. 4.

3 In Exercises 5 6, evaluate the given expression with and Let. Find all scalars k such that. 8. Let. Find all scalars k such that. In Exercises 9 10, find and In Exercises 11 12, find the Euclidean distance between u and v. 11.

4 Find the cosine of the angle between the vectors in each part of Exercise 11, and then state whether the angle is acute, obtuse, or 90. ; θ is acute ; θ is obtuse ; θ is obtuse 14. Find the cosine of the angle between the vectors in each part of Exercise 12, and then state whether the angle is acute, obtuse, or Suppose that a vector a in the xy-plane has a length of 9 units and points in a direction that is 120 counterclockwise from the positive x-axis, and a vector b in that plane has a length of 5 units and points in the positive y-direction. Find. 16. Suppose that a vector a in the xy-plane points in a direction that is 47 counterclockwise from the positive x-axis, and a vector b in that plane points in a direction that is 43 clockwise from the positive x-axis. What can you say about the value of? In Exercises 17 18, determine whether the expression makes sense mathematically. If not, explain why. 17.

5 does not make sense because is a scalar. makes sense. does not make sense because the quantity inside the norm is a scalar. makes sense since the terms are both scalars Find a unit vector that has the same direction as the given vector. 20. Find a unit vector that is oppositely directed to the given vector. 21. State a procedure for finding a vector of a specified length m that points in the same direction as a given vector v. 22. If and, what are the largest and smallest values possible for? Give a geometric explanation of your results. 23. Find the cosine of the angle θ between u and v.

6 24. Find the radian measure of the angle θ (with ) between u and v. and and and and In Exercises 25 26, verify that the Cauchy-Schwarz inequality holds Let and. Describe the set of all points for which. A sphere of radius 1 centered at. 28. Show that the components of the vector in Figure Ex-28a are and. Let u and v be the vectors in Figure Ex-28b. Use the result in part to find the components of.

7 29. Prove parts and of Theorem Prove parts and of Theorem Prove parts and (e) of Theorem Figure Ex Under what conditions will the triangle inequality (Theorem 3.2.5a) be an equality? Explain your answer geometrically. 33. What can you say about two nonzero vectors, u and v, that satisfy the equation? 34. What relationship must hold for the point to be equidistant from the origin and the xz-plane? Make sure that the relationship you state is valid for positive and negative values of a, b, and c. What relationship must hold for the point to be farther from the origin than from the xz-plane? Make sure that the relationship you state is valid for positive and negative values of a, b, and c -False Exercises In parts (j) determine whether the statement is true or false, and justify your answer. If each component of a vector in is doubled, the norm of that vector is doubled. In, the vectors of norm 5 whose initial points are at the origin have terminal points lying on a circle of radius 5 centered at the origin. Every vector in has a positive norm. False

8 If v is a nonzero vector in, there are exactly two unit vectors that are parallel to v. (e) If,, and, then the angle between u and v is radians. (f) The expressions and are both meaningful and equal to each other. False (g) If, then. False (h) If, then either or. False (i) In, if u lies in the first quadrant and v lies in the third quadrant, then cannot be positive. (j) For all vectors u, v, and w in, we have Copyright 2010 John Wiley & Sons, Inc. All rights reserved.