Final answer:
The curve for r(t) = (t, 2 - t, 2t) is sketched by plotting points in three-dimensional space based on the value of t at each coordinate. As t increases, the path moves in a diagonal direction, with the x-coordinate increasing, y-coordinate decreasing, and z-coordinate increasing. An arrow indicates the increasing direction of t.
Step-by-step explanation:
To sketch the curve with the given vector equation r(t) = (t, 2 - t, 2t), we must analyze each component of the vector as a function of t. The vector components correspond to coordinates in three-dimensional space, where t is a parameter that represents time or another continuous variable.
To indicate with an arrow the direction in which t increases, we will sketch the path that the vector follows as t varies and add an arrow at a point where the value of t is higher than at the starting point.
Starting at t = 0, we have the point (0, 2, 0). As t increases, the x-coordinate increases, the y-coordinate decreases, and the z-coordinate increases. For example, at t = 1, the point would be (1, 1, 2). Thus, the curve will proceed diagonally in the positive direction of the x-axis, negative direction of the y-axis, and positive direction of the z-axis, forming a three-dimensional curve.
We can further illustrate this by graphically representing each positional vector for a series of increments in t and connecting these points smoothly with a continuous line, ensuring to draw an arrow at the end of the line to indicate the direction of increasing t.