Tangent Vectors and Normal Vectors. Definitions of Unit Tangent Vector.

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Presentation transcript:

Tangent Vectors and Normal Vectors

Definitions of Unit Tangent Vector

1. Find the unit tangent vector to the curve at the specified value of the parameter (Similar to p.865 #5-10)

2. Find the unit tangent vector to the curve at the specified value of the parameter (Similar to p.865 #5-10)

Note Given a tangent vector T(t) at a point: (x 1, y 1, z 1 ): Form: k Direction numbers: a = a 1, b = b 1, c = c 1 Parametric Equations: x = at + x 1, y = bt + y 1, z = ct + z 1

3. Find the unit tangent vector T(t) and find a set of parametric equations for the line tangent to the space curve at point P (Similar to p.865 #11-16)

4. Find the unit tangent vector T(t) and find a set of parametric equations for the line tangent to the space curve at point P (Similar to p.865 #11-16)

5. Find the unit tangent vector T(t) and find a set of parametric equations for the line tangent to the space curve at point P (Similar to p.865 #11-16)

Definition: Principal Unit Normal Vector

6. Find the principal unit normal vector to the curve at the specified value of the parameter (Similar to p.866 #23-30)

7. Find the principal unit normal vector to the curve at the specified value of the parameter (Similar to p.866 #23-30)

8. Find v(t), a(t), T(t), and N(t) (if it exists) for the an object moving along the path given by the vector-valued function r(t) (Similar to p.866 #31-34)

Tangential and Normal Components of Acceleration

9. Find T(t), N(t), a T, and a N at the given time t for the plane curve r(t) (Similar to p.866 #35-44)

10. Find T(t), N(t), a T, and a N at the given time t for the plane curve r(t) (Similar to p.866 #35-44)

11. Sketch the graph of the plane curve given by the vector-valued function, and, at the point on the curve determined by r(t o ), sketch the vectors T and N. Note that N points toward the concave side of the curve (Similar to p.866 #49-54)

12. Sketch the graph of the plane curve given by the vector-valued function, and, at the point on the curve determined by r(t o ), sketch the vectors T and N. Note that N points toward the concave side of the curve (Similar to p.866 #49-54)

13. Find T(t), N(t), a T, and a N at the given time t for the space curve r(t) [Hint: Find a(t), T(t), a T, and a N. Solve for N in the equation a(t) = a T T + a N N.] (Similar to p.866 #55-62)