18 02A Problem Set 5 due in class on Thurs Oct 23 Part I 26 points TB Simmons SN 18 02A Supplementary Notes all have solutions The problems marked other are not to be handed in Topic 17 R Oct 16 Vectors dot product Read TB 17 3 18 1 18 2 Hand in 1A 3a 4a 7bc 8ab 11 1B 1a 2a 3a 5b 11 13 Others 1A 1 2 6 Topic 18 M Oct 20 Determinants cross product Read SN D TB 18 3 Hand in 1C 2b 3b 4 9 1D 1b 2 5 6 Others 1C 1 5 1D 1a 4 Topic 19 T Oct 21 Matrices inverses Read SN M 1 M 2 Hand in 1F 5b 8a 1G 3 4 5 Others 1F 3 Continuation W Oct 22 Discussion review and catch up Coming next Topic 20 R Oct 23 pset 5 due Square matrices systems Cramer s rule planes Read SN M 3 M 4 Problem section F Oct 25 Topic 21 M Oct 28 Parametric equations and vector derivatives Read TB 17 1 17 2 to middle of page 598 18 4 Part II 24 points Directions Try each problem alone for 20 minutes If you collaborate later you must write up solutions independently Problem 1 Class 17 4 pts A man traveling east at a speed U finds that the wind seems to blow directly from the north On doubling his speed he finds it appears to come from the northeast Find the velocity speed and direction of the wind Problem 2 Class 17 6 pts 2 4 In this problem we examine tacking which is the process sailboats use to travel against the wind Sails are a familiar tool to harness the energy of the wind for transportation over the sea Early ships had large fixed sails which would capture the wind blowing from behind to 1 18 02A Problem Set 5 2 propel the ship forward Even if the wind is blowing from behind at an acute angle the component of the wind vector perpendicular to the sail will push on the sail and hence on the boat However these early fixed sail ships had no way to go against the wind and had to rely on oarsmen if the wind was blowing in the wrong direction A great advance that allowed boats to sail against the wind was the invention of movable sails in combination with a rudder and a keel By carefully positioning the sail the boat can be made to sail into the wind this process is called tacking As noted before the component of the wind perpendicular to the sail pushes on the sail and through it the boat The keel only allows the boat to move along its axis The rudder is used to turn the boat That is for any force on the boat only the component along the boat s axis actually pushes the boat Described mathematically the wind vector is first projected on the perpendicular to the sail to get the direction of the force on the sail This resultant force is projected on the axis of the boat to find the direction the boat is being pushed By orienting the sail correctly this double projection can result in a vector with a component pointing into the wind The two pictures show the same thing w ai is the force of the wind on the sail the line lS is parallel to the sail with 0 2 and the line lB is along the the boat s axis with 0 2 y y lB lB ls ls w ai x w ai x a Let w1 be the component of w perpendicular to the line lS Give a sketch showing that w1 points rightward as does w b Find the projection of w1 onto lB Give an explicit formula in terms of and What is the condition on and that this projection has a component in the i direction Problem 3 Class 18 6 pts 3 3 a Pick any vertex of a cube and find the cosine of the angle between the main diagonal from that vertex and one of the adjacent face diagonals from the same vertex b Find the cosine of the angle between the two main diagonals Problem 4 Class 18 4 pts 2 2 The Biot Savart law which you will study in 8 02 says that the magnetic field B at a point x y z with position vector r xi yj zk induced by a current of magnitude J passing 18 02A Problem Set 5 through the origin in the k direction is given by B constant and r r 3 0 k r where 0 is a physical J 4 r3 a Find the formula for the magnetic field B at a general point x y z and evaluate it at the point 1 2 3 b For which points x y z at unit distance from the origin is the magnitude of the magnetic field B largest Problem 5 Class 19 4 pts 1 3 pts 1 2 0 Consider the matrix 0 1 1 4 4 2 a Compute det A b Compute A 1 by computing in turn the following matrices i minors ii cofactors iii adjoint then dividing by det A Check by matrix multiplication the the result found satisfies AA 1 I End of pset 5
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