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1. Vectors and Matrices - 1. Vectors and Matrices 1A....

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1. Vectors and Matrices

1. Vectors and Matrices - 1. Vectors and Matrices 1A....

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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 (5 T) that A makes respectively with i , j , and k . a) Show that dir A = cos cr i + cos Pj + cos y k . (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.
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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.
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