Week 1 Quiz – MATH 163

Week 1 Quiz – MATH 163 – Spring 2021 Make sure to show all of your work. Please follow the guidelines for “writing math well” (see pdf link and examples in the Getting Started Section). You may use your own textbook and your own notes. You may not consult with other humans. All work should be your own. Any indication that you have copied or shared your work will result in zero points on this assessment. All of your quiz work MUST be submitted by 11:59pm on Friday, April 9. Problem 1: In this problem you will be creating a sphere shape in ℝ3 a) b) c) d) Pick an ordered pair that is NOT the origin. This will be the center of your sphere. Pick a radius for your sphere. This radius may NOT be zero. Write the equation for your sphere using the information you chose in parts a and b. Use Mathematica to graph part c. Include a 3D axes in your picture, similar to the pictures we made in class. Problem 2: For this problem, create two vectors. Both vectors should be in ℝ3 and they should be different vectors. Both vectors must have a magnitude bigger than 0. I encourage you to keep your entries relatively small integer values. You may use positive or negative entries (or a mixture). a) Write down your two vectors in component form. Call one of them 𝑣⃑ and one of them 𝑤 ⃑⃑⃑. b) Compute 2𝑣⃑ + 𝑤 ⃑⃑⃑. Write your final answer in terms of the standard basis vectors. c) Compute the magnitude of your vector 𝑣⃑ d) Create the unit vector in the same direction as your vector 𝑤 ⃑⃑⃑ 1 e) Explain, using words, how the vector − 𝑤 ⃑⃑⃑ is different from 𝑤 ⃑⃑⃑. 2 Problem 3: Hand draw a set of coordinate axes in ℝ3 . Label the axes so that they do NOT follow the right hand rule. Then explain using words what you would need to change about your picture so that it would follow the right hand rule. H ARVEY M UDD C OLLEGE D EPARTMENT OF M ATHEMATICS WRITING MATHEMATICS WELL Communicating mathematics well is an important part of doing mathematics. As you write up your homework solutions, keep these things in mind: • Write in sentences. Complete thoughts are sentences that end in periods. You may still highlight important equations by displaying them, but even displayed equations should have punctuation! Use paragraphs to separate important ideas. • Use helpful connective phrases. “If”, “then”, “so”, “therefore”, “we see that”, “recall that”, … • Your audience is other students in the class who have not seen this problem before. Remind the reader of any relevant facts from class or the book. Your solution should give adequate detail so that the reader can follow your solution. • It is possible to write too much! If you write out every triviality, the reader may get lost in the details. This is not good writing, either. (In particular, really trivial calculations need not be shown.) • Avoid shorthand. Don’t use arrows, and write out ’for all’, ’there exists’. • You may wish to outline your problem-solving strategy at the beginning of the problem. Example. Here are two different solutions to the same problem. Which one is easier to understand? (0 − 3)2 + (x − 2)2 = 25 32 = 9 + (x2 − 4x + 4) = 25 x2 − 4x − 12 (x − 6)(x + 2) =⇒ x = −2, 6 x > 0 x=6 WHY THIS IS POORLY WRITTEN: • You don’t know what problem the writer was solving. • You can’t tell what’s an assumption and what’s a conclusion. • Where does one thought end and another begin? There are no sentences! • In the 2nd line: combining two thoughts can create untruths (32 is 9 but it isn’t 25). • The 3rd line dangles; what’s being asserted here? It’s not a sentence. • What’s the relationship between all these phrases? Connective phrases would help! Problem. Find a point in the plane on the positive x-axis that has distance 5 from the point (2, 3). Solution. The desired point is (6, 0). To find this, we note if (x, 0) is a solution, then x must must satisfy the equation (x − 2)2 + (0 − 3)2 = 25, which follows from the planar distance formula between the points (x, 0) and (2, 3). It follows that x2 − 4x + 13 = 25. Then x2 − 4x − 12 = 0. Factoring, we obtain (x − 6)(x + 2) = 0, satisfied by either x = −2 or x = 6. Since we assumed x > 0 and y = 0, we see (6, 0) is the desired point. WHY THIS IS WELL-WRITTEN: • The writer described the problem, and strategy for solution. • Every thought is a complete sentence with subject and verb (the “equals” sign is a verb). • She answered the question right at the beginning. (Boxing answers is customary.) • Notice even the equations have punctuation (comma, periods) as they are part of sentences. • She highlighted important ingredients, displayed important equations, avoided trivial algebra. Writing well will benefit you, too! It helps you structure your own thinking, and you will thank yourself when you re-read your solutions later.
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