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Vectors and Parameterization

​A recurring strand in Exeter's math curriculum

Vectors and parametric representations of points arise in a variety of settings throughout our problem sets. Here are a few examples.

Math 1: From its initial position at (−1, 12), a bug crawls linearly with constant speed and direction. It passes (2, 8) after two seconds. How much time does the bug spend in the first quadrant?

(Math 1, #872)

Math 2: A direction vector for a line is any vector that joins two points on that line. Find a direction vector for 2x + 5y = 8. It is not certain that you and your classmates will get exactly the same answer. How should your answers be related, however?

(Math 2, 20#8)

Math 3: Graph the curve that is described parametrically by (xy) = (5cost, 4sint). If you are working in degree mode, the parameter interval should of course be 0 ≤ t ≤ 360. This presentation should remind you of the parametric description of a circle. The curve is actually an ellipse. Confirm this by substituting the parametric equations into the ellipse equation 16x2 + 25y2 = 400.

(Math 3, 46#7)

Math 4: The curve (xy) = (t2t3 − t) is shown at right. Find the area enclosed by the loop in the graph.

(Math 4, 74#9)

Other strands that arise throughout our problem sets: