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midterm 2006(2)

# midterm 2006(2) - HKUST MATH 102 Second Midterm Examination...

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HKUST MATH 102 Second Midterm Examination Multivariable and Vector Calculus 15 Dec 2006 Answer ALL 8 questions Time allowed – 180 minutes Problem 1 (a) Find an equation of the plane through ( - 1 , 4 , - 3) and perpendicular to the line x = t + 2 , y = 2 t - 3 , z = - t. (b) Find a rectangular equation for the surface whose spherical equation is ρ = 2 sin θ sin φ . De- scribe the surface. (c) Show that the two lines r = a + v t and r = b + u t , where t is a parameter and a , b , u and v are constant vectors, will intersect if ( a - b ) · ( u × v ) = 0. Problem2 (a) Describe the graph of the equation r 1 ( t ) = - 2 i + t j + ( t 2 - 1) k . Find also the vector equation of the tangent line to the curve r 1 ( t ) such that it is parallel to the line r 2 ( t ) = i + (2 + 2 t ) j + (3 + 4 t ) k . (b) Sketch the surfaces x + y = 4 and y 2 4 2 + z 2 2 2 = 1 in the first octant. Find the parametric equations of the curve C of intersection of the two surfaces above. Find the parametric equation of the projection curve C onto the xz -plane. Describe the projection curve.

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midterm 2006(2) - HKUST MATH 102 Second Midterm Examination...

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