Final Answer:
The vector function r(t) = (6t, 2t², 4t³).
Step-by-step explanation:
The vector function r(t) describes a parametric curve in three-dimensional space. It represents a path traced by a point as the parameter t varies. In this case, the function r(t) is defined as (6t, 2t², 4t³).
Breaking down the components of r(t):
The x-component of the vector is 6t.
The y-component of the vector is 2t².
The z-component of the vector is 4t³.
Each component of the vector function is dependent on the parameter t. As t changes, the point represented by the vector r(t) moves along the curve described by the function.
For instance, when t = 1:
x-component = 6 * 1 = 6
y-component = 2 * 1² = 2
z-component = 4 * 1³ = 4
Therefore, the point on the curve described by r(t) when t = 1 is (6, 2, 4). Similarly, for different values of t, the corresponding points on the curve can be obtained by substituting the value of t into the vector function.
The curve traced by the vector function r(t) represents a specific path in three-dimensional space. The x, y, and z components of the vector function correspond to the coordinates of points along this path as the parameter t changes. This function can be used to model various physical phenomena or describe the motion of objects in space, providing a mathematical representation of their positions at different times.