Question

Imagine that you are driving on a twisting mountain road. Describe the unit tangent vector, principal unit normal vector, and binomial vector as you ascend, descend, twist right, and twist left.

Short answer

The unit tangent vector, the normal vector, and the binomial vector form a right-hand triple. So, when the road is bending to the left it points up, and when the road bends to the right, it points down.

Step-by-step solution

Step 1. Given Information.

It is given that we're driving on a twisting mountain road.

Step 2. Describe.

The unit tangent vector straights in the direction of motion of the road. The principal unit normal vector is orthogonal to the unit tangent vector and points into the road. The plane in which the road fits best at the point of tangency is the osculating plane which is determined by the unit tangent vector. The binomial vector is normal to the osculating plane. Thus. the unit tangent vector, the normal vector, and the binomial vector form a right-hand triple. So, when the road is bending to the left it points up, and when the road bends to the right, it points down.