Explain how you would determine how much error there is between a vector addition and the displacement of the actual trip taken. Why aren’t vectors true straight - line paths in the real world? Would you expect the vector addition to be larger or smaller than the actual trip taken? Why?

Explanation:

No doubt your curriculum materials have an adequate discussion of error and how it is determined.

a) The error between two values is the difference between an approximation and the true value. You determine it by subtracting the true value from the approximation.

In the case of a trip, you need to decide what is the "true value" and what is the "approximation." The wording here suggests that the "displacement of the actual trip taken" is to be considered the "true value." Then, you determine the error by subtracting that displacement from the "vector addition."

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b) In Euclidean geometry, a vector is a straight-line path. In the real world, we use the term vector to refer to the direction taken by an object, generally along a curved path at some relative distance from the Earth's surface. The direction reference may change along the path, so that following a given "vector" will usually result in a curved path (not a great circle) on the Earth's surface.

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c) The curved path from a point A to a point B on the Earth's surface will always be longer than the straight-line path between the same two points. In the case of points on opposite sides of the world, the straight-line path is through the center of the Earth, whereas the "vector addition" will be some path along the surface of the Earth.

You need to decide the meaning of "actual trip taken," as any actual trip will generally involve all the changes in direction and ups and downs along the way. If you consider "the actual trip taken" to be the difference between starting and ending coordinates, then the "vector addition" will always be longer.

the area between the graph and the t axis