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Exam1 Review - 10 11 MAT 397 REVIEW 1 You must clearly show...

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Unformatted text preview: 10. 11. MAT 397 REVIEW 1 9-25-2007 You must clearly show all the steps in your arguments to receive credit. . Find the direction cosines of the vector b 2 2i — 3j + k. Write b 2 2i +j — k as the sum of two vectors, one parallel to a : i —j — k and the other orthogonal to a. Find the vector, parametric and symmetric forms of the equation of the line through the point P(2, 4, —1) orthogonal to the plane a: — 2g + 32: = 4. Consider the three points P(1,0, 1), Q(2, —1, 2) and R(-—1,0, —1). (a) Find the equation of the plane containing these points. (b) Find the area of the triangle determined by these points. Find the equation of the plane through the point P(—1, 2,0) and perpendicular to the vector from the origin to P. Find the equation of the line of intersection of the planes 2x + 23/ — z = 4 and ac — y ‘ z = 2. The two lines a: =1+t,y 2 215,2: 2 —3t and a: = 2+s,y :1+s,z = —2(1 +3) intersect. (a) Find their point of intersection. (b) Find the equation of the plane which contains these two lines. Consider the surface given in cylindrical coordinates by z = r2. Sketch the surface and find it’s equation in rectangular coordinates. Sketch the curve r(t) = tsin ti + tcos tj + tk, indicating the direction of increasing t. Consider the curve r(t) = eti + et sin tj + 6‘ cos tk. (a) Find a unit tangent vector and unit normal to the curve when t = 0. (b) Find the equation of the osculating plane to the curve when t = 0. Assume the position vector of a particle is given by r(t) = sin ti + tj + cos tk. (a) Find it’s velocity, speed and acceleration at time t = 0. (b) How far does the particle travel during the time interval 0 S t g 1'? ...
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