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Vector Calculus Independent Study

Unit 1 Sample Test

1. An astronomer observes a comet at position (7, −3, 5) in a coordinate system with the Earth at the origin (and distances measured in astronomical units, or aus).

(a) [5 points] If the Sun is at position (−1, 0, 0), what is the distance between the Sun and the comet?

(b) [15 points] The Earth, the Sun, and the comet form a triangle. What is the area of this triangle? [10 points] What is the equation of the plane that it lies on?

(c) [10 points] There is an unique line going through the Sun and the Comet. What is the parametric description of this line? [20 points] What is the projection of the Earth onto this line? (Your answer should be a point in 3 space).

2. [10 points] Show that (2, 1, 6), (5, 7, 9), and (8, 5, −6) are the vertices of a right triangle. [10 points] Find the area of this triangle.

3. [20 points] Convert the equation x + z = y^{2} to both cylindrical and spherical coordinates.