Vectors Motion

We can generalize the description of average velocity from one to many dimensions by noting that, for orthogonal (i.e. perpendicular) directions, the description of the position as a vector and the rules for vector addition (and hence vector subtraction as well) extend the definition for average velocity quite naturally.

Consider the figure below that shows the position of an object as initially r₀ and, some time later, r_f.

Then the velocity is

with Dt being the time between the initial and final position determinations.

The utility of a vector description for multi-dimensional motion makes it easy to do seemingly complicated problems. For example, suppose we are asked to solve the following problem: a boat wants to cross a river in which the stream flows at a velocity v_s. The crossing is to be done such that the boat goes straight across from point A to point B as shown in the figure below.

The boat has a speed v_b. The question is: what heading the boat should take to affect the shortest passage (i.e. one directly from point A to point B)?

Solution:
The fact that velocities are describable as vectors mean that they take on a central property of vectors, namely the principle of superposition. This principle states that the resultant velocity of an object is simply the vector sum of the velocities due to all things acting on the object. In this case, the resultant velocity of the boat, v_r , is due to the vector sum

Thus we can solve the problem by noting that we want a heading q such that the x components of the velocities for the boat's velocity in still waters (v_{b, x} ) and the stream's velocity cancel (since the stream velocity is all along the x direction). Therefore, we want

Since we want a straight path from point A to point B, we demand that

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