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 2 i j 2 k 1A 2 For what value s of c will i c 3 i 6 j 2 k j c k be a unit vector 1A 3 a If P 1 3 1 and Q 0 1 I find A P Q A and dir A b A vector A has magnitude 6 and direction i 2 0 I where is its head 2j 2 k 3 If its tail is a t 1A 4 a Let P and Q be two points in space and X the midpoint of the line segment P Q Let 0 be an arbitrary fixed point show that as vectors O X OP OQ b With the notation of part a assume that X divides the line segment P Q in the ratio r s where r s 1 Derive an expression for O X in terms of O P 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 c k 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 called the direction cosines of A cos P j cos y k The three coefficients are b Express the direction cosines of A in terms of a b c ofthevector i 2 j 2 k find the direction cosines 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 v 2 1 E 18 02 EXERCISES 2 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 parallelogram 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 P R 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 c i a be orthogonal 2 j k and i j 2k will 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 value s of a will PQR be a a right angle Q 0 1 I R For what b an acute angle 1B 5 Find the component of the force F 2 i 2j a the direction a 1 3 j 4 k in b the direction of the vector 3 i 2j 6 k 1 VECTORS AND MATRICES 3 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 j 1B 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 3 j 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 1B 8 The vectors i i j k A j x fi kt i j 2k 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 …
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