1. Vectors and Matrices 1A. Vectors Definition. A direction is just a unit vector. The direction of A is defined by A dir A = -, (A # 0);IAI it is the unit vector lying along A and pointed like A (not like -A). 1A-1 Find the magnitude and direction (see the definition above) of the vectors a) i+j+k b) 2i-j+2k c) 3i-6j-2k 1A-2 For what value(s) of c will $ i -j + ck be a unit vector? 1A-3 a) If P = (1,3, -1) and Q = (0,1, I),find A = PQ, (A(, and dir A. b) A vector A has magnitude 6 and direction (i + 2j -2 k)/3. If its tail is at (-2,0, I), where is its head? 1A-4 a) Let P and Q be two points in space, and X the midpoint of the line segment PQ. Let 0 be an arbitrary fixed point; show that as vectors, OX = $(OP + OQ) . b) With the notation of part (a), assume that X divides the line segment PQ in the ratio r : s, where r + s = 1. Derive an expression for OX in terms of OP and OQ. 1A-5 What are the i j -components of a plane vector A of length 3, if it makes an angle of 30' with i and 60' with j . Is the second condition redundant? 1A-6 A small plane wishes to fly due north at 200 mph (as seen from the ground), in a wind blowing from the northeast at 50 mph. Tell with what vector velocity in the air it should travel (give the i j -components). 1A-7 Let A = a i + b j be a plane vector; find in terms of a and b the vectors A' and A" resulting from rotating A by 90' a) clockwise b) counterclockwise. (Hint: make A ttie diagonal of a rectangle with sides on the x and y-axes, and rotate the whole rectangle.) c) Let i' = (3 i + 4j)/5. Show that i' is a unit vector, and use the first part of the exercise to find a vector j ' such that i', j ' forms a right-handed coordinate system. 1A-8 The direction (see definition above) of a space vector is in engineering practice often given by its direction cosines. To describe these, let A = a i + b j + ck be a space vector, represented as an origin vector, and let a, p, and y be the three angles (5T)that A makes respectively with i , j ,and k . a) Show that dir A = cos cr i + cos Pj + cos yk . (The three coefficients are called the direction cosines of A.) b) Express the direction cosines of A in terms of a, b, c; find the direction cosines ofthevector -i +2j +2k. c) Prove that three numbers t, u, v are the direction cosines of a vector in space if and only if they satisfy t2 + u2+ v2 = 1.2 E. 18.02 EXERCISES 1A-9 Prove using vector methods (without components) that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length. (Call the two sides A and B.) 1A-10 Prove using vector methods (without components) that the midpoints of the sides of a space quadrilateral form a parallelogram. 1A-11 Prove using vector methods (without components) that the diagonals of a parallel- ogram bisect each other. (One way: let X and Y be the midpoints of the two diagonals; show X = Y .) 1A-12* Label the four vertices of a parallelogram in counterclockwise order as OPQR. Prove that the line segment from 0 to the midpoint of PQ intersects the diagonal PR in a point X that is 113 of the way from P to R. (Let A =OP, and B = OR; express everything in terms of A and B.) 1A-13* a) Take a triangle PQR in the plane; prove that as vectors PQ +QR +RP = 0. b) Continuing part a), let A be a vector the same length as PQ, but perpendicular to it, and pointing outside the triangle. Using similar vectors B and C for the other two sides, prove that A + B + C = 0. (This only takes one sentence, and no computation.) 1A-14* Generalize parts a) and b) of the previous exercise to a closed polygon in the plane which doesn't cross itself (i.e., one whose interior is a single region); label its vertices pi, P2,. .. ,Pn as you walk around it. 1A-15* Let Pi,. . . ,P, be the vertices of a regular n-gon in the plane, and 0 its center; show without computation or coordinates that OPl + OP2+ ... + OPn = 0, a) if n is even; b) if n is odd. 1B. Dot Product 1B-1 Find the angle between the vectors 1B-2 Tell for what values of c the vectors ci + 2j -k and i -j + 2 k will a) be orthogonal b) form an acute angle 1B-3 Using vectors, find the angle between a longest diagonal PQ of a cube, and a) a diagonal PR of one of its faces; b) an edge PS of the cube. (Choose a size and position for the cube that makes calculation easiest.) 1B-4 Three points in space are P : (a, 1,-l), Q : (0,1, I), R : (a, -1,3). For what value(s) of a will PQR be a) a right angle b) an acute angle ? 1B-5 Find the component of the force F = 2 i -2j + k in a) the direction + j -b) the direction of the vector 3 i + 2j -6 k .43 1. VECTORS AND MATRICES 1B-6 Let 0 be the origin, c a given number, and u a given direction (i.e., a unit vector). Describe geometrically the locus of all points P in space that satisfy the vector equation In particular, tell for what value(s) of c the locus will be a) a plane b) a ray (i.e., a half-line) c) empty (Hint: divide through by (OPI.) i + j1B-7 a) Verify that i' = -and j' = -j are perpendicular unit vectors that -+fi fi form a right-handed coordinate system b) Express the vector A = 2 i -3j in the i'j'-system by using the dot product. c) Do b) a different way, by solving for i and j in terms of i' and j' and then substituting into the expression for A. i+j+k j'x . . i+j-2k1B-8 The vectors i'= A ' fi kt= are three mutually perpendicular unit vectors that form a right-handed coordinate system. a) Verify this. b) Express A = 2 i + 2j-k in this system (cf. 1B-7b) 1B-9 Let A and B be two plane vectors, neither one of which is a multiple of the other. Express B as the sum of two vectors, one a multiple of A, and the other perpendicular to A; give the answer in terms of A and B. (Hint: let u = dir A; what's the u-component of B?) 1B-10 Prove using vector methods …
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