How Far Will You Travel? Portfolio PRECALCULUS: VECTORS Directions: 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 at surface. However, when nding 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 a, to three-dimensional vectors in order to nd the shortest distance between two A points on the surface of Earth. Hopefully you remember from Unit 1 Lesson 3 how we calculated arc length from the radius of a circle and the central angle in radians (s = r6). We will use this relationship in this portfolio to find the distance between two cities. To do this, we will approximate the earth as a sphere with radius, r, and 9 will be the angle formed between the vectors from the earth's center to each of the cities. To find the angle, 6, between the vectors to each city, we will use the dot product and the magnitude of the vectors. We have been doing this in 2-dimensions in lessons 5 and 6 of our current unit. In this portfolio we will do it in 3-dimensions. The dot product and magnitude are found the same way in 2 or 3 dimensions. Given two 3 dimensional vectors, u = and v = c Dot product is: u - v = (x1)( x2) + (y1)(y2) + (21)( 22) o Magnitude is: ||u|| =\ fxi + y: + zi and "VII =/x: + y: + 2: Now putting it all together we can find the angle between the vectors just like we did in lesson 6 in 2 dimensions. u-v = ||u||||v||cosEl 11'1} HHIIHVII c059 = 3 = cos_1("u1|1|-I1|:v") Let's try it and see how accurately we can nd the distance between Vancouver and Newport, Maine! Every point on Earth can be represented by a three-dimensional vector. The vector's starting point is at the center of Earth. To calculate the unit vectors corresponding to each of your locations, apply these formulas: v = (x, y, z), where the following applies: x = cos(Latitude)-cos(Longitude) y = 603(Latitude)-sin(Longitude) z = sin(Latitude) The latitude and longitude for Vancouver and New Port in degrees are: _ Vancouver, British Columbia 49.2827" N 123.1207" W Newport, Maine 44.8371 N 69.2853" W ***Use a Negative sign for southern latitudes or western longitudes ***Use a Positive sign for northern latitudes or eastern longitudes. Part 1: Calculate the corresponding three - dimensional unit vector for each location. 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 in calculating each coordinate. Use 5 decimal places for all calculations! (3 pts) Location x- coordinate y- coordinate z- coordinate cos(latitude)*cos(longitude) cos(latitude)*sin(longitude) sin(latitude) Vancouver, British Columbia Newport, Maine Part 2: Find the angle 0 between your 2 vectors. Show your work and final answer. Make sure your final angle 0 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). Remember your angle needs to be in radians when you compute arc length. Although Earth is not a perfect sphere, assume that it is for the purposes of this portfolio and use the formula for the arc length of a segment of a great circle on a sphere, s = r9, to nd the distance between your two locations. In this case, r is the radius of Earth, which is approximately 3,963.2 miles. Find the distance between your two locations. Part 4: Write a paragraph (at least 3 sentences) 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. Consider why your online source might be less or more accurate than your calculation (2 points)