Question

Sketch the graph of the plane curve given by the vector - valued function, and, at the point on the curve determined by r (t0), sketch the vectors T and N. Note that N points toward the concave side of the curve.

Function                                                                 Time

Answer 1

Step 1 :

If is the position vector, then

the unit tangent vector is .

If tangent vector is , then

the principal unit normal vector is or .

Step 2 :

The vector-valued function is and time .

Apply derivative on each side with respect to t.

Find the magnitude :

Step 3 :

Find the tangent vector :

Substitute in above equation.

Compare the above with standard form of tangent vector.

and .

Substitute and in or

.

or .

Step 4 :

Graph :

Graph the parametric equations and the normal vector .

Solution :

or .

Graph :