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Math 150 – Fall 2015 Chapter 8 & 9 Review 1 of 1 Chapter 8 & 9 Review Formulas: For the new material (chapters 8 & 9) these are the formulas you need to MEMORIZE: π • sin−1 x: Domain [−1, 1], Range [ −π 2 , 2] • cos−1 x : Domain [−1, 1], Range [0, π] π • tan−1 x. Domain (−∞, ∞), Range ( −π 2 , 2) • Law of Sines: sin A a = sin B b = sin C c • Law of Cosines: c2 = a2 + b2 − 2ab cos C • 2-dimensions: ~i = h1, 0i, ~j = h0, 1i • 3-dimensions: ~i = h1, 0, 0i, ~j = h0, 1, 0i, ~k = h0, 0, 1i p • Vector length ||hx1 , x2 , . . . , xn i|| = x21 + x22 + · · · + x2n • Dot Product: hx1 , x2 , . . . , xn i · hy1 , y2 , . . . , yn i = x1 y1 + x2 y2 + · · · xn yn • Angle between vectors: ~x · ~y = ||x|| ||y|| cos θ or cos θ = ~x · ~y ||~x|| ||~y || Note. These are the formulas you need to memorize for Chapter 8 & 9, but you also need to know how to solve the problems! Example 1. Find a unit vector which points in the same direction of the vector from the point P (−3, 5) to the point Q(0, −1). Example 2. Determine the angle between the vectors h−2, 4i and h3, 6i. Example 3. Suppose a man walks 100 yards in a direction 30◦ north of east. Assuming that the origin is the man’s starting point and that east is the positive x-axis, what are the coordinates of the man’s final location? Example 4. Find the length of the vector from the point P (2, 1, −5) to the point Q(0, −3, 2).