Math32a 1 R Kozhan Midterm 1 Jan 30 2012 Name UID Circle your TA and discussion session 1A Tues Anthony Ruozzi 1C Tues Xiaokui Yang 1B Thur Anthony Ruozzi 1D Thur Xiaokui Yang Instructions If you get stuck move on to the next question You don t have a lot of time Show all work if you want to get full credit You have an extra scratch paper on the last page If you need more raise your hand and ask No books notes electronics incl calculators and cell phones are allowed Good luck and have fun Question Max Your score 1 12 2 7 3 11 4 10 5 10 Total 50 1 Problem 1 a 2 points Find the distance from the point 2 3 4 to the y axis b 2 points Let b h2 1 2i Find a vector of length 5 in the direction opposite to b c 2 points Let a h1 1 0i b h2 1 2i Find a unit vector orthogonal to both a and b d 2 points Let a h1 1 0i b h2 1 2i Find the vector projection proj b a e 2 points Do planes 2x 8y 6z 2 and x 1 4y 3z intersect Explain f 2 points Write the definition of the vector triple product of a b c 2 Problem 2 7 points Let A 4 2 and B 1 1 be two given points in R2 Consider the set of all points P with coordinates x y in R2 such that the distance from P to A is twice the distance from P to B Show that this set is a circle and find its center and radius Hint what s the formula for the distance between two points 3 Problem 3 3 4 Suppose that a b are vectors in R3 such that the length of a is 3 the length of b is 2 and the angle between them is Note choosing some specific a and b satisfying the above conditions and computing the expressions below will earn you 3 points out of 11 You need to compute these expressions for any choice of such a and b a 3 points Find a b a 2 b b 5 points Find the length of a b a 2 b c 3 points Find a 3 b 3 a b 4 Problem 4 Consider two points in R3 A 1 3 0 and B 2 2 2 a 3 points Find an equation of the line passing through the points A and B b 4 points Find the coordinates of the point where this line intersects the plane P given by the equation x y z 1 c 3 points Does this plane P intersect the line segment between A and B Briefly explain your answer no credit if your solution contains only yes or no 5 Problem 5 Consider a particle with the trajectory given by the following curve called an ellipse pictured on the right x 4 cos 3t y 2 sin 3t a 1 point Assuming t 0 is the initial time when is the next time the particle return to its original starting point b 3 points Find the slope of the tangent line at time t known number 9 it s okay to leave the answer as a trig function of a c 6 points Compute the area of the interior of this ellipse 5 points for the correct final formula 1 point for final computation Hint you might want to consider splitting the curve into pieces Note if you already know the formula for the area of ellipse you can t use it zero credit Some of the following formulae could be useful sin 2 2 1 cos 2 cos2 2 1 cos 2 sin2 2 sin cos 6 Extra scratch paper 7
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