Tangent Vector

     We expect the tangent vector at to be the limit vector which is the vector derivative. To find parametric equations of the tangent line at we use the point corresponding to and the tangent vector



to obtain parametric equations

(1) Proposition (Tangent Vector) Suppose is differentiable at and that Then is a tangent vector to the graph of at the point where and points in the direction of increasing .

(2) Example (Tangent Vector) Find a tangent vector at the point where for



    Solution. We have,



and the tangent line to the graph of for is the line that passes through the point and is determined by the parametric equations    and because this line passes through and is parallel to the tangent vector at namely,

(3) Definition (Smooth Curve) The graph of the vector function defined by is smooth on any interval of where is continuous and .

(4) Example (Smooth Curve) Determine where the graph of the vector function



is smooth.

    Solution. The graph of the vector function is smooth over any interval not containing because



and for any except

    If the graph of a vector function is a smooth curve then the nonzero derivative is tangent to at the point that corresponds to ; and in this case we can make since of the following,

(5) Definition (Motion of an Object in Space) If an object moves in such a way that its position at time is given by the vector function whose graph is a smooth curve then

    (i)  Velocity vector:  
    
    (ii)  Speed  
    
    (iii)  Direction of motion vector:  
    
    (iv)  Acceleration vector:  
    

(6) Example (Motion of an Object in Space) Suppose the position vector for a particle in space at time is given by Find the particle's velocity vector, acceleration vector, speed, and direction of motion vector at time

    Solution. At time , the particle's velocity vector is



the acceleration vector is,



the speed is



and the direction of motion vector is

.

(7) Example (Motion of an Object in Space) Find the position vector and velocity vector given the acceleration vector function initial position vector   and initial velocity vector  

    Solution.  Given the velocity vector function is
     




where and are constants to be determined. By using



Thus Therefore,



So the position vector function is



which is



where and are constants to be determined. By using



Thus Therefore,

(8) Definition (Unit Tangent and Unit Normal Vectors) If is a vector function that defines a smooth graph, then at each point a unit tangent vector is



and the principal unit normal vector function is

    Suppose an object moves along the smooth graph of a vector function in the direction of increasing Then at each point on the trajectory the unit tangent points in the direction of motion, while the principal unit normal vector points in the direction the object is turning. For trajectory in a plane his means that points toward the concave side of the trajectory.   

(9) Example (Unit Tangent and Unit Normal Vectors) Given



find the unit tangent vector and principal unit normal vector

    Solution. We have,











and

(10) Proposition (Arc Length Function) Let be a piecewise smooth curve that is the graph of the vector function described parametrically by ,   and let be a particular point on (base point). Then the length of from the base point to the variable is given by the arc length function

(11) Example (Arc Length Function) Find the arc length of the curve



from to

    Solution.  We have, units because







and so the arc length is

(12) Proposition (Speed as the Derivative of Arc Length) Suppose an object moves along a smooth curve that is the graph of the position function , where is continuous on the interval Then the object has speed for where  

(13) Example (Speed as the Derivative of Arc Length) If a moving object has a position vector function of



then find the speed of the object at time and the distance traveled by the object between times and

    Solution. The speed of the object at time is   because



  

and the distance traveled by the object between times and is because

   .
  

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