Vector Basics , 7, 1 2, 1 2

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1 Vector Basics 1. Sketch each vector, assuming the initial point is at the origin. (a) 3, 4 (b) 2, 8 (c) 5, 0, 4 2. Calculate the magnitude of each of the vectors in Question The initial point P and terminal point Q of a vector are given below. Sketch each vector and write it in component form. (a) P (3, 1), Q(7, 2) (b) P (5, 1, 2), Q(5, 8, 0) 4. Find the sum of the given vectors and illustrate geometrically. (a) v = 1, 4, w = 8, 4 (b) v = 3, 6, w = 7, 2 (c) v = 9, 1 3, w = 0, For each pair of vectors v, w in Question 4, sketch the vectors v, w, and v w. Calculate the difference v w and check it against your drawing. 6. Prove each of the following statements for three-dimensional vectors u, v, w and scalars (that is, real numbers) a, b. (a) u + v = v + u (vector addition is commutative) (b) u + (v + w) = (u + v) + w (vector addition is associative) (c) a(u + v) = au + av (scalar multiplication is distributive over vector addition) (d) (a + b)u = au + bu (e) au = a u 7. Let u = 3, 4 and v = 1, 1. Find scalars s and t so that the given equation is satisfied: (a) su + tv = 6, 0. (b) su + 8, 11 = tv. 8. Which of the following are unit vectors? 2 1 5, 5, 3 1 7, 7, 1 2, 1 2. Normalize those that are not already unit vectors. 9. Find a vector that has the same direction as 2i + 4j but has length Two nonzero vectors u and v are said to be linearly independent in the plane if they are non parallel. (a) If u and v have this property and au = bv for constants a, b, show that a = b = 0. (b) Show that the standard representation of a vector is unique. That is, if the vector u has the representation u = a 1 i + b 1 j and u = a 2 i + b 2 j is another such representation, then a 1 = a 2 and b 1 = b Suppose u and v are a pair of nonzero, nonparallel vectors. Find the set of real triples (a, b, c) such that au + b(u v) + c(u + v) = 0. Does it matter whether the vectors u and v are two-dimensional or three-dimensional? 1

2 Geometry problems using vectors 1. (a) Let three collinear points A, B, C have position vectors a, b, c respectively and suppose AC CB = λ µa+λb µ. Show that c = µ+λ. (b) Let points A, B have position vectors a, b. Using the part (a), show that the midpoint of AB has position vector 1 2 (a + b). (c) Use vectors to find the coordinates of the midpoint of the line segment joining the points P ( 3, 8) and Q(9, 2). (d) What point is located 5 6 of the distance from P to Q? 2. Show that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half of its length. 3. Show that the diagonals of a parallelogram bisect each other. 4. In a triangle, let u, v, w be the vectors from each vertex to the midpoint of the opposite side. Show that u + v + w = Let the position vectors of the vertices of a triangle be a, b, and c. Prove that the medians of the triangle intersect at a single point using vector methods and find the position vector of this point (called the centroid) in terms of a, b, and c. 6. Let A, B, C, D be the vertices of a quadrilateral and let M, N, P, Q be the midpoints of the sides AB, BC, CD, AD respectively. Show that MNP Q is a parallelogram. 7. A spy finds some map coordinates, a crude drawing, and these words: From the snowman, pace off the distance to the palm tree, turn left, pace off an equal distance, then drive a stake. From the snowman again, pace off the distance to the cactus, turn right, pace off an equal distance, and drive a second stake. Dig halfway between the stakes. The spy arrives at the location indicated by the map coordinates. There is a palm tree and a cactus, but the snowman has melted! How can the spy know where to dig? 8. The head pirate of the Bermuda triangle wants to find the location of a sunken treasure ship and dive down to get the gold. He knows the GPS coordinates of four points, and also some instructions on how to find the ship: From point A, sail half of the distance to point B, then a third of the distance to point C, and finally a fourth of the distance to point D. Unfortunately, the pirate does not know how the coordinates are labeled, so he does not know which point to start from. What is the minimum number of dives he must make in order to find the treasure? 9. Prove that the lines joining the midpoints of the opposite edges of a tetrahedron are concurrent and bisect each other. 2

3 Dot Product Problems 1. For u = 1, 2, 3, v = 1, 0, 1, and w = 0, 2, 1, compute (a) (u + v) w (b) 2v + 4w (c) u w + v w 2. Check that i i = j j = k k = 1 and also that i j = j k = k i = Prove the following for three-dimensional vectors u, v, w and scalar k. (a) u v = v u (b) u (v + w) = u v + u w (c) k(u v) = (ku) v = u (kv) 4. Determine whether the given vectors are orthogonal, parallel, or neither. (a) u = 5, 3, 7, v = 6, 8, 2 (b) u = 4, 6, v = 3, 2 (c) u = i + 2j + 5k, v = 3i + 4j k 5. What is the projection of v = 5, 0 onto u = 3, 4? 6. The Triangle Inequality for vectors is a + b a + b. (a) Give a geometric interpretation of the Triangle Inequality. (b) Prove the Triangle Inequality. 7. Let AB be the diameter of a circle, and let C be any other point on the circumference. Show that ABC is a right triangle, with right angle at C. 8. Give the vector equation of the line passing through the points A(1, 4, 2) and B(5, 9, 6). 9. Give the vector equation of the plane passing through the points Q(4, 1, 2), R(1, 0, 2), and S(3, 1, 4). 3

4 Extra Problems 1. Use vectors to decide whether the triangle with vertices P (1, 3, 2), Q(2, 0, 4), and R(6, 2, 5) is right-angled. 2. For what values of b are the vectors 6, b, 2 and b, b 2, b orthogonal? 3. Let a = a 1,..., a n be a vector in R n. Show that if x = x 1,..., x n and x = x 1,..., x n are the position vectors of two points that satisfy n a i x i = b, i=1 for some real number b, then the vector y from one point to the other satisfies a y = (a) Find the angle between a diagonal of a cube and one of its edges. (b) Find the angle between a diagonal of a cube and a diagonal of one of its faces. 5. If c = a b + b a, where a, b, and c are all nonzero vectors, show that c bisects the angle between a and b. 6. Prove the following inequality: a b a b. 7. The Parallelogram Law states that a + b 2 + a b 2 = 2 a b 2 (a) Give a geometric interpretation of the Parallelogram Law. (b) Prove the Parallelogram Law. 8. A line segment AB, where A is a point on plane P, is said to be perpendicular to P if it is perpendicular to every line on the plane through A. A famous hard theorem in Euclid (hard to prove by the methods available to Euclid), is: If AB is perpendicular to two different lines through A in P, then it is perpendicular to P. Prove this theorem using dot products. 9. Prove that the altitudes of a triangle intersect at a common point. (An altitude of a triangle is a straight line through a vertex and perpendicular to the opposite side.) 10. If r = x, y, z, a = a 1, a 2, a 3, and b = b 1, b 2, b 3, show that the vector equation (r a) (r b) = 0 represents a sphere, and find its center and radius. 4

5 Cross Product Problems 1. Calculate: i j, j k, k i, i i, j j, k k. 2. Find the cross product a b: (a) a = 1, 2, 0, b = 0, 3, 1 (b) a = t, t 2, t 3, b = 1, 2t, 3t 2 (c) a = 3i + 2j + 4k, b = i 2j 3k. 3. Find two unit vectors orthogonal to both 2, 0, 3 and 1, 4, Find the area of the parallelogram with vertices A( 2, 1), B(0, 4), C(4, 2), D(2, 1). 5. Use the scalar triple product to verify that the vectors u = i + 5j 2k, v = 3i j, and w = 5i + 9j 4k are coplanar. 6. Determine whether the points A(1, 3, 2), B(3, 1, 6), C(5, 2, 0), and D(3, 6, 4) lie in the same plane. 7. Find the volume of the parallelepiped determined by the vectors a = 6, 3, 1, b = 0, 1, 2, c = 4, 2, Let a, b, and c be three-dimensional vectors. Suppose that a 0. (a) If a b = a c, does it follow that b = c? (b) If a b = a c, does it follow that b = c? (c) If a b = a c and a b = a c, does it follow that b = c? 5

6 Extra Cross Product Problems 1. Let v = 5j and let u be a vector with length 3 that starts at the origin and rotates in the xy-plane. Find the maximum and minimum values of the length of the vector u v. In what direction does u v point? 2. Find the area of the parallelogram with vertices K(1, 2, 3), L(1, 3, 6), M(3, 8, 6), and N(3, 7, 3). 3. Let a, b, and c be three-dimensional vectors. Prove that (a b) (a + b) = 2(a b). 4. Let a, b, and c be three-dimensional vectors. Prove the following formula for the vector triple product: a (b c) = (a c)b (a b)c. 5. Let a, b, and c be three-dimensional vectors. Use the previous question to prove that a (b c) + b (c a) + c (a b) = Let a, b, and c be three-dimensional vectors. Prove that (a b) (c d) = (a c)(b d) (a d)(b c). 7. A tetrahedron is a solid with four vertices, P, Q, R, and S, and four triangular faces. (a) Let v 1, v 2, v 3, and v 4 be vectors with lengths equal to the areas of the faces opposite the vertices P, Q, R, and S respectively, and directions perpendicular to the respective faces and pointing outward. Show that v 1 + v 2 + v 3 + v 4 = 0. (b) The volume V of a tetrahedron is one-third the distance from a vertex to the opposite face, times the area of that face. Find a formula for the volume of a tetrahedron in terms of the coordinates of its vertices P, Q, R, and S. (c) Suppose the tetrahedron in the figure has a trirectangular vertex S. (This means that the three angles at S are all right angles.) Let A, B, and C be the areas of the three faces that meet at S, and let D be the area of the opposite face P QR. Using the results of part (a), or otherwise, prove that D 2 = A 2 + B 2 + C 2 (This is a three-dimensional version of the Pythagorean Theorem.) 6