VECTOR DATA IS ESSENTIAL

When two people agree to exchange information on directions, they must inherently agree to use vectors. You can describe most trips by small straight line directions. In other words, a collection of straight lines forms a good approximation to a curvy trip. Still, detailing the trip requires many pieces of information. For example, to parametrize trips with time, requires a direction and a velocity of travel at each time, t, during the entire trip. Mathematize for giving directions along the map of World Wallpaper would have the following form. At time t the vector (v_1(t),v_2(t)) has length \sqrt{(v_1(t))^2+ (v_2(t))^2}, showing the velocity of travel of the person. The travel direction follows a line slope v_2(t)/v_1(t).

Question D.1: If this data on giving directions is so important, why don't people use vectors in real life interactions? [Adam, this could be a question to the student. We could ask them to discuss their feelings on it. The next discussion could be an answer we provide for those who would like to hear what we have to say after they have responded.

Suppose a direction giver holds out his arm and points in the direction of travel. This pointing arm represents a real life version of a vector on paper. Using the phrase tangent vector requires understanding to what the vector is tangent. When giving directions, tangent vectors are tangent to the path of travel. That is, they are parallel to lines showing direction along the curve of travel. In real curve motion, direction is constantly changing. Thus, a tangent vector is the direction of travel at a given moment in time. That is, one can stop moving along the curve and look at the tangent vector to see the actual direction of travel.

People familiar with the destination and route of travel often forget they rely on directions. Yet, aren't they following a set of subconscious directions? Whenever a person moves, he or she follows directions using vectors. Even a simple trip from the bedroom to the kitchen requires the brain to tell muscles which direction to walk. When scientists monitor activities of a dynamical system, they require data for the movement of many pieces of the system. This data must allow them to reconstruct the activities of the system parts. With even a few particles moving, data for significant motion can be complicated. Pictorial representation helps when significant events have the most pronounced movements attached to them.

In many scientific events, however, significant motions build over time from tiny motions. Consider an example. Suppose one builds a bridge and the bridge sways causing a certain pole holding two pieces together to bend too quickly for its tolerance. You wouldn't see this in a picture: Such bending might be the result of differences between two large movements. Still, eventually, the bridge might fail because of this movement.

Question D.2: Since High School students train so much on length and slope of lines, why is it valuable to switch from slope ideas to vector ideas. [Question the student participating in the Interactive Presentation. For example, does the student think slope and length is easier than vectors. A picture might represent a path of a particle's travel, with 10, say, points along it for the participant to click at. When they do click there, a help figure pops up with two pieces of data in it. 1) A vector based at the origin appears. It has length equal to the velocity of the particle at the point; it points in the direction of the particles travel. 2) A box encloses the pair (velocity, slope) for the point. This should show, even in the plane of World Wallpaper, the vector picture is visually simpler than the numerical data.]

Representing data with vectors may seem only slightly easier in the plane than using the data of slope and velocity. When, however, we consider motion in 3-space, the extra complication requires vectors.


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