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Math 230 — Summer 2023

Mastery Assignment 1

Due June 3, 2023 11:59 PM

Instructions:

• Answers to these problems must uploaded to GradeScope using this template (if possible). Otherwise, you must map your answers to the right question in GradeScope at the time of upload. Answers must be organized and legible. Points may be deducted if anything is unclear about the process of your work.

• Points will be deducted for incomplete reasoning and disorganized work (even if your answers are correct).

• If you have any difficulties/questions, you are encouraged to discuss these problems with me.

• Late assignments must go into the Late Assignment Dropbox on Canvas.

1. (5 points) Particle 1 starts at point (3, 1, 1) and particle 2 starts at point (10, 5, −4); at t = 0, both particles begin to move along linear paths. After 1 second, particle 1 is at point (4, 3, 4) and particle 2 is at (8, 6, 3).

(a) Do the paths of the particles intersect? Show all work.

(b) Will the particles collide? Show all work.

2. (5 points) A plane flies with velocity v = h220, −90, 10i km/h. A wind is blowing out of the northeast with velocity w = h−30, −30, 0i km/h.

(a) Express the wind vector as a sum of vectors, one parallel to the plane’s velocity, one perpendicular to it.

(b) Is the parallel part of the wind blowing with or against the plane?

(c) If we account for both v and w, what is the direction vector that describes the actual motion of the plane?

3. (5 points) Consider the tetrahedron (a 4-sided pyramid whose faces are all triangles) with vertices at A = (1, 0, 0), B = (0, 2, 0), C = (0, 1, 3), and D = (2, 2, 0). Calculate the surface area and the volume of the tetrahedron.(Hint: You may use the fact that the volume a tetrahedron is exactly one-sixth of the volume of the parallelepiped generated by the same vectors.)

4. (5 points) Find the parametrization of the curve defined by the intersection of the two surfaces y = x2 − 3z2 and x2 + z2 = 9, where x ≥ 0.