The equation r(t) = sin(4t)i + cos(4t)j, 0t≥0 describes the motion of a particle moving along the unit circle. Answer the following questions about the behavior of the particle. a. Does the particle have constant speed? If so, what is its constant speed?
b. Is the particle's acceleration vector always orthogonal to its velocity vector?
c. Does the particle move clockwise or counterclockwise around the circle?
d. Does the particle begin at the point (1,0)?

a) Particle has a constant speed of 4, b) Velocity and acceleration vector are orthogonal to each other, c) Clockwise, d) False, the particle begin at the point (0,1).

Step-by-step explanation:

a) Let is find first the velocity vector by differentiation:

Where the resultant vector is the product of a unit vector and magnitude of the velocity vector (speed). Velocity vector has a constant speed only if magnitude of unit vector is constant in time. That is:

Then,

Hence, the particle has a constant speed of 4.

b) The acceleration vector is obtained by deriving the velocity vector.

Velocity and acceleration are orthogonal to each other only if . Then,

Which demonstrates the orthogonality between velocity and acceleration vectors.

c) The particle is rotating clockwise as right-hand rule is applied to model vectors in 2 and 3 dimensions, which are associated with positive angles for position vector. That is:

And cosine decrease and sine increase inasmuch as t becomes bigger.

d) Let evaluate the vector in .

False, the particle begin at the point (0,1).