Preview Activity 10.6.1.
All your good work as CEO and lead engineer at Steer Clear has started to pay off, literally. When you showed your work on using the location tracking system (LTS) to develop navigation and telemetry tools to an investment group (your grandparents), they were impressed and decided to give you money to further develop your self-driving car. You decided to use this infusion of cash to buy a gyroscopic accelerometer which will measure acceleration as a vector with magnitude and direction. In order to use your new instrument for information on the “driving” part of your self-driving car, you need to understand how your accelerometer readings will relate to the operations needed to drive a car.
(a)
The image below shows the path you drove on a test drive along with points \(P\) and \(Q\) and the direction of travel. At the points \(P\) and \(Q\text{,}\) we have draw three vectors. The acceleration vector \(\va\) provided by your new gyroscopic accelerometer is shown in magenta. The two blue vectors are the unit tangent vector \(\vT\) and unit normal vector \(\vN\) at that point. Label each of the blue vectors as being \(\vT\) or \(\vN\text{.}\)
A two-dimensional curve with two points indicated. At each point, there are three vectors.
(b)
Draw the projection of \(\va\) on \(\vT\) and the projection of \(\va\) on \(\vN\) at point \(P\) and at point \(Q\) in the graph above.
(c)
Is \(\va \cdot \vT\) is positive, negative, or zero at \(P\text{?}\) What about at \(Q\text{?}\) Write a sentence or two to justify your answers.
(d)
Is \(\va \cdot \vN\) is positive, negative, or zero at \(P\text{?}\) What about at \(Q\text{?}\) Write a sentence or two to justify your answers.
