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Fall 2010 Math 152 2 Week in Review XIV Section 13.4 Key Concepts: courtesy: David J. Manuel 1. Polar to Cartesian: x = r cos θ, y = r sin θ (covering 11.3 and 13.4) 2. Cartesian to Polar: r2 = x2 + y 2 , tan θ = 1 3. Graphing Polar Curves (use quadrants and symmetry) Section 11.3 Key Concepts: 1. Cross Product (|a||b| sin θ)n y x Examples: definition: a × b = 1. Convert the following points to cartesian coordinates: π (a) 4, 3 √ π (b) 2 2, − 4 4π (c) 6, − 3 2. Computation formula: a×b = (a2 b3 −a3 b2 )i+(a3 b1 −a1 b3 )j+(a1 b2 − a2 b1 )k 3. Result: a vector orthogonal to the plane containing a and b whose magnitude is |a||b| sin θ (area of the parallelogram formed by a and b) 2. Convert the following points to polar coordinates: 4. Scalar Triple Product = |a · (b × c)| = volume of the parallelipiped formed with a, b and c at one corner. (a) (−3, 3) √ (b) (−2 3, −2) √ √ (c) ( 6, − 2) Examples: 3. Sketch the graphs of the following polar equations: 1. Find a × b if a = h1, −1, 2i and b = j − 3k. 2. Find a vector parallel to the plane containing the points A(2, 0, 1), B(4, 3, −1) and C(3, 2, 3). (a) r = 3 cos θ (b) r2 = cos 2θ 1 (c) r = (1 − 2 sin θ) 2 (d) r = a(1 + cos θ) 3. Find the area of △ABC in #2 4. Find the volume of the parallelipiped which has a corner formed by the vectors h−4, 3, −1i, h2, 3, −2i, and h1, 2, 2i. 4. Convert r = 3 cos θ + 2 sin θ to cartesian and sketch the graph. 5. Find a unit vector orthogonal to the vectors −i + j + 3k and −2j + k. 6. A 0.75m pipe wrench is held to a connecting bolt at an angle of 30◦ with the x-y plane. Because of space limitation, the force of 20N must be applied straight down (i.e., perpendicular to the x-y plane). Find the magnitude and direction of the torque. 1