Name Date How Far Will You Travel? Worksheet
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 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. Part 1
Choose your starting location and a dream destination location and find the latitude
and longitude for each.
• Round to four decimal places. • Use a negative sign for southern latitudes and/or western longitudes, and
use a positive sign for northern latitudes and/or eastern longitudes. Name of Location Latitude Longitude Cite your source where you found the latitude and longitude for both locations: © 2016 Connections Education LLC. All rights reserved. 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) â‹… cos(longitude)
=
y cos(latitude) â‹… sin(longitude) z = sin(latitude)
Calculate the corresponding three-dimensional unit vector for each of your
locations. Name of Location x-coordinate y-coordinate z-coordinate © 2016 Connections Education LLC. All rights reserved. x, y , z 2 Part 2
In order to find the distance between these two locations, you need to know the
angle between the two vectors (or the central angle). To find the angle, you will
need to find the dot product of your two vectors and find the magnitude of each.
Finding the dot product of 3D vectors is essentially the same as finding the dot
product of 2D vectors: a, b, c d , e, f = ad + be + cf
Finding the magnitude of 3D vectors is also essentially the same as it is for 2D
vectors: a , b, c = a 2 + b2 + c2 Now find the angle θ between your two vectors using this formula: vw v w θ = cos −1 Show your work in the space below. Be sure to find θ in radians, not in degrees: © 2016 Connections Education LLC. All rights reserved. 3 Part 3
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 = rθ, to find 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.
Show your work in the space below. What is the distance between your two locations? Part 4
Write a paragraph that discusses what you learned about vectors from this
portfolio. © 2016 Connections Education LLC. All rights reserved. 4

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