How Far Will You Travel Portfolio 2 and 3 Dimensional Vectors Questions

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Suppose that you plan to take a trip to your dream destination. You would like to know the shortest distance between your starting point and your destination. When calculating distances on a plane, you need only consider two dimensions because you are on a flat surface. However, when finding distances between two points on earth, you must take into the account the curvature of a sphere. In this portfolio, you will extend your knowledge of two-dimensional vectors to three-dimensional vectors in order to find the shortest distance between two points on the surface of earth.

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Mrs. Spray and Dr. H are providing you with data to complete this portfolio as they are the most amazing, wonderful Precalculus Teachers! So this part is done for you! Use data below for your calculations in the portfolio. Name of Location Miami Florida Anchorage, Alaska Latitude 25.7671 o N 61.2181o N Use a Negative sign for southern latitudes or western longitudes Use a Positive sign for northern latitudes or eastern longitudes. Longitude 80.1918o W 149.9003o W Part 1: Calculate the corresponding three - dimensional unit vector for each location. As usual you must show your work to earn credit. If you just enter values in the table you will not get credit! Show the equation and values used for calculating each coordinate. Use 5 decimal places in all your calculations! ( 3 points) Location Miami, Florida x- coordinate y- coordinate z- coordinate Anchorage, Alaska Part 2: Find the angle θ between your 2 vectors. Show your work and final answer. Make sure your angle θ is in radians not degrees. Use 5 decimal places in all your calculations. ( 4 points) Part 3: Find the distance between the two locations to the nearest mile. Show your work and final answer with units. (2 points) Part 4: Write a paragraph that discusses what you learned about vectors from this portfolio. (1 point) Part 5: Look up the actual distance between the two cities (include your source). Calculate the percentage of error in your calculation and comment on how accurate it is and possible sources of error. (2 points)
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Explanation & Answer

Attached you will find the answers of your task in Word and PDF format. The equations were made using the Microsoft Office Equation Editor. Please let me know if you need edits 😊

Miami, Florida
𝑥 = cos(25.7671°) × cos(−80.1918°) = +0.15341
𝑦 = cos(25.7671°) × sin(−80.918°) = −0.88741
𝑧 = sin(25.7671°) = +0.43471

〈𝑥, 𝑦, 𝑧〉 = 〈+0.15341, −0.88741, +0.43471〉 = 〈𝑎, 𝑏, 𝑐〉

Anchorage, Alaska
𝑥 = cos(61.2181°) × cos(−149.9003°) = −0.41655
𝑦 = cos(61.2181°) × sin(−149.9003°) = −0.24146
𝑧 = sin(61.2181°) = +0.87646

〈𝑥, 𝑦, 𝑧〉 = 〈−0.41655, −0.24146, +0.87646〉 = 〈𝑑, 𝑒, 𝑓〉

‖〈𝑎, 𝑏, 𝑐〉‖ = √𝑎2 + 𝑏 2 + 𝑐 2 = √(0.15341)2 + (−0.88741)2 + (+0.43471)2 = 1.00000
‖〈𝑑, 𝑒, 𝑓〉‖ = √𝑑...

Awesome! Perfect study aid.


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