Final answer:
The vector equation r(t) = 4 cos(t)i + 4 sin(t)j + k corresponds to a helical path in a 3D space. The helix circles in the xy-plane with a constant upward motion in the z-direction. An arrow indicates the direction of increasing t, showing the motion of the curve.
Step-by-step explanation:
The student is asked to sketch the curve of the given vector equation r(t) = 4 cos(t)i + 4 sin(t)j + k. This equation represents a circular helix with a radius of 4 and a constant k value in the z-direction, meaning the circle moves upwards in a helical pattern as t increases. The parameter t represents time or a parameter that modifies the vector's position.
Sketching this curve involves plotting the components of r(t) in a three-dimensional coordinate system. The i and j components 4 cos(t) and 4 sin(t) respectively represent the x and y components that form a circle of radius 4 in the xy-plane. The k component means that for each value of t, there is a constant increase in the z-direction, which gives the helical shape.
To indicate the direction in which t increases and therefore show the motion of the particle, an arrow is drawn tangent to the curve pointing in the direction that the curve is traversing as t increases.