The position of a particle in motion in the plane at time t is
X(t) = −7 t I + sin(−3 t) J.
At time any t, determine the following:
the unit tangent vector to X(t) is: ??? I + ??? J
(Note that, a unit vector is a vector whose length is 1. Multiplying any non-zero vector V by 1/|V| produces a unit vector, and multiplying any unit vector by -1 shows that there are two unit vector which are multiples of any non-zero vector. The unit tangent vector to X(t) is defined as V(t)/|V(t)|. )

Question

The position of a particle in motion in the plane at time t is
X(t) = −7 t   I + sin(−3 t)   J.
At time any t, determine the following:
the unit tangent vector to X(t) is: ??? I + ??? J
(Note that, a unit vector is a vector whose length is 1. Multiplying any non-zero vector V by 1/|V| produces a unit vector, and multiplying any unit vector by -1 shows that there are two unit vector which are multiples of any non-zero vector. The unit tangent vector to X(t) is defined as V(t)/|V(t)|. )

anonymous · Answer

r'(t)=-7i-3cos(-3t)
|r'(t)|=$$\sqrt{49+9\cos^2(-3t)}$$

-7/\sqrt{49+9\cos^2(-3t)}i-3\cos(-3t)/\sqrt{49+9cos^2(-3t)k

anonymous · Answer

-7/sqrt{49+9cos^2(-3t)}i-3cos(-3t)/sqrt{49+9cos^2(-3t)k