Linear Algebra/Length and Angle Measures/Solutions

Solutions Problem 1 Find the length of each vector. <li> <li> <li> <li> <li> Answer <li> <li> <li> <li> <li> Problem 2 Find the angle between each two, if it is defined. <li> <li> <li> Answer <li> <li> <li> Not defined. Problem 3 During maneuvers preceding the Battle of Jutland, the British battle cruiser Lion moved as follows (in nautical miles): miles north, miles degrees east of south, miles at degrees east of north, and miles at degrees east of north. Find the distance between starting and ending positions O'Hanian 1985. Answer We express each displacement as a vector (rounded to one decimal place because that's the accuracy of the problem's statement) and add to find the total displacement (ignoring the curvature of the earth). The distance is . Problem 4 Find so that these two vectors are perpendicular. Answer Solve to get . Problem 5 Describe the set of vectors in orthogonal to this one. Answer The set can also be described with parameters in this way. Problem 6 <li> Find the angle between the diagonal of the unit square in and one of the axes. <li> Find the angle between the diagonal of the unit cube in and one of the axes. <li> Find the angle between the diagonal of the unit cube in and one of the axes. <li> What is the limit, as goes to , of the angle between the diagonal of the unit cube in and one of the axes? Answer <li> We can use the -axis. <li> Again, use the -axis. <li> The -axis worked before and it will work again. <li> Using the formula from the prior item, . Problem 7 Is any vector perpendicular to itself? Answer Clearly is zero if and only if each is zero. So only is perpendicular to itself. Problem 8exer:AlgPropsDotProd Describe the algebraic properties of dot product. <li> Is it right-distributive over addition: ? <li> Is is left-distributive (over addition)? <li> Does it commute? <li> Associate? <li> How does it interact with scalar multiplication? As always, any assertion must be backed by either a proof or an example. Answer Assume that have components . <li> Dot product is right-distributive. <li> Dot product is also left distributive: . The proof is just like the prior one. <li> Dot product commutes. <li> Because is a scalar, not a vector, the expression makes no sense; the dot product of a scalar and a vector is not defined. <li> This is a vague question so it has many answers. Some are (1) and , (2) (in general; an example is easy to produce), and (3) (the connection between norm and dot product is that the square of the norm is the dot product of a vector with itself). Problem 9exer:EqOfCauchySchwarz Verify the equality condition in Corollary 2.6, the Cauchy-Schwarz Inequality. <li> Show that if is a negative scalar multiple of then and are less than or equal to zero. <li> Show that if and only if one vector is a scalar multiple of the other. Answer <li> Verifying that for and is easy. Now, for and , if then , which is times a nonnegative real. The half is similar (actually, taking the in this paragraph to be the reciprocal of the above gives that we need only worry about the case). <li> We first consider the case. From the Triangle Inequality we know that if and only if one vector is a nonnegative scalar multiple of the other. But that's all we need because the first part of this exercise shows that, in a context where the dot product of the two vectors is positive, the two statements "one vector is a scalar multiple of the other" and "one vector is a nonnegative scalar multiple of the other", are equivalent. We finish by considering the case. Because and , we have that . Now the prior paragraph applies to give that one of the two vectors and is a scalar multiple of the other. But that's equivalent to the assertion that one of the two vectors and is a scalar multiple of the other, as desired. Problem 10 Suppose that and . Must ? Answer No. These give an example. Problem 11 Does any vector have length zero except a zero vector? (If "yes", produce an example. If "no", prove it.) Answer We prove that a vector has length zero if and only if all its components are zero. Let have components . Recall that the square of any real number is greater than or equal to zero, with equality only when that real is zero. Thus is a sum of numbers greater than or equal to zero, and so is itself greater than or equal to zero, with equality if and only if each is zero. Hence if and only if all the components of are zero. Problem 12 Find the midpoint of the line segment connecting with in . Generalize to . Answer We can easily check that is on the line connecting the two, and is equidistant from both. The generalization is obvious. Problem 13 Show that if then has length one. What if ? Answer Assume that has components . If then we have this. If then is not defined. Problem 14 Show that if then is times as long as . What if ? Answer For the first question, assume that and , take the root, and factor. For the second question, the result is times as long, but it points in the opposite direction in that . Problem 15 unit vector A vector of length one is a unit vector. Show that the dot product of two unit vectors has absolute value less than or equal to one. Can "less than" happen? Can "equal to"? Answer Assume that both have length . Apply Cauchy-Schwarz: . To see that "less than" can happen, in take and note that . For "equal to", note that . Problem 16 Prove that Answer Write and then this computation works. Problem 17 Show that if for every then . Answer We will prove this demonstrating that the contrapositive statement holds: if then there is a with . Assume that . If then it has a nonzero component, say the -th one . But the vector that is all zeroes except for a one in component gives . (A slicker proof just considers .) Problem 18 Is ? If it is true then it would generalize the Triangle Inequality. Answer Yes; we can prove this by induction. Assume that the vectors are in some . Clearly the statement applies to one vector. The Triangle Inequality is this statement applied to two vectors. For an inductive step assume the statement is true for or fewer vectors. Then this follows by the Triangle Inequality for two vectors. Now the inductive hypothesis, applied to the first summand on the right, gives that as less than or equal to . Problem 19 What is the ratio between the sides in the Cauchy-Schwarz inequality? Answer By definition where is the angle between the vectors. Thus the ratio is . Problem 20 Why is the zero vector defined to be perpendicular to every vector? Answer So that the statement "vectors are orthogonal iff their dot product is zero" has no exceptions. Problem 21 Describe the angle between two vectors in . Answer The angle between and is found (for ) with If or is zero then the angle is radians. Otherwise, if and are of opposite signs then the angle is radians, else the angle is zero radians. Problem 22 Give a simple necessary and sufficient condition to determine whether the angle between two vectors is acute, right, or obtuse. Answer The angle between and is acute if , is right if , and is obtuse if . That's because, in the formula for the angle, the denominator is never negative. Problem 23 Generalize to the converse of the Pythagorean Theorem, that if and are perpendicular then . Answer Suppose that . If and are perpendicular then (the third equality holds because ). Problem 24 Show that if and only if and are perpendicular. Give an example in . Answer Where , the vectors and are perpendicular if and only if , which shows that those two are perpendicular if and only if . That holds if and only if . Problem 25 Show that if a vector is perpendicular to each of two others then it is perpendicular to each vector in the plane they generate. (Remark. They could generate a degenerate plane— a line or a point— but the statement remains true.) Answer Suppose is perpendicular to both and . Then, for any we have this. Problem 26 Prove that, where are nonzero vectors, the vector bisects the angle between them. Illustrate in . Answer We will show something more general: if for , then bisects the angle between and x150px (we ignore the case where and are the zero vector). The case is easy. For the rest, by the definition of angle, we will be done if we show this. But distributing inside each expression gives and , so the two are equal. Problem 27 Verify that the definition of angle is dimensionally correct: (1) if then the cosine of the angle between and equals the cosine of the angle between and , and (2) if then the cosine of the angle between and is the negative of the cosine of the angle between and . Answer We can show the two statements together. Let , write and calculate. Problem 28 Show that the inner product operation is linear: for and , . Answer Let and then as required. Problem 29arithmetic geometric mean inequality The geometric mean of two positive reals is . It is analogous to the arithmetic mean . Use the Cauchy-Schwarz inequality to show that the geometric mean of any is less than or equal to the arithmetic mean. Answer For , set so that the Cauchy-Schwarz inequality asserts that (after squaring) as desired. ? Problem 30 A ship is sailing with speed and direction ; the wind blows apparently (judging by the vane on the mast) in the direction of a vector ; on changing the direction and speed of the ship from to the apparent wind is in the direction of a vector . Find the vector velocity of the wind Ivanoff Esty 1933. Answer This is how the answer was given in the cited source. The actual velocity of the wind is the sum of the ship's velocity and the apparent velocity of the wind. Without loss of generality we may assume and to be unit vectors, and may write where and are undetermined scalars. Take the dot product first by and then by to obtain Multiply the second by , subtract the result from the first, and find Substituting in the original displayed equation, we get Problem 31 Verify the Cauchy-Schwarz inequality by first proving Lagrange's identity: and then noting that the final term is positive. (Recall the meaning and of the notation.) This result is an improvement over Cauchy-Schwarz because it gives a formula for the difference between the two sides. Interpret that difference in . Answer We use induction on . In the base case the identity reduces to and clearly holds. For the inductive step assume that the formula holds for the , ..., cases. We will show that it then holds in the case. Start with the right-hand side and apply the inductive hypothesis to derive the left-hand side. References O'Hanian Hans Physics 1 W. W. Norton 1985 Ivanoff V. F. (proposer) Esty T. C. (solver) Problem 3529 American Mathematical Monthly 39 2 February 1933 118