Question

Sketch the curve represented by the vectorvalued function and give the orientation of the curve. \(\mathbf{r}(t)=t^{2} \mathbf{i}+2 t \mathbf{j}+\frac{3}{2} t \mathbf{k}\)

Short answer

The curve is upward opening parabola in the three-dimensional coordinate system, and the orientation of the curve is in the direction in which \( t \) increases, i.e., it moves from left to right as we increase the \( x \)-coordinate and goes upwards with increasing \( y \)- and \( z \)-coordinates.

Step-by-step solution

Understand the components

First of all, understand the three functions that compose the vector-valued function \( \bold{r}(t) = t^2\bold{i}+2t\bold{j}+\frac{3}{2}t\bold{k} \). Here, \( \bold{i} \), \( \bold{j} \), and \( \bold{k} \) denote the unit vectors along the \( x \), \( y \), and \( z \) axes respectively. Therefore, \( t^2 \), \( 2t \), and \( \frac{3}{2}t \) are the \( x \), \( y \), and \( z \) coordinates respectively of the curve as a function of \( t \).

Plot the components

Plot the three functions \( x = t^2 \), \( y = 2t \), and \( z = \frac{3}{2}t \) separately on their respective two-dimensional planes. Note the curve trends: \( x = t^2 \) is a parabolic curve which opens upwards, \( y = 2t \) is a straight line with slope 2, and \( z = \frac{3}{2}t \) is a straight line with slope \frac{3}{2}, all as functions of \( t \).

Combine the components

Combine the three components to plot the curve in a three-dimensional coordinate system. For every value of \( t \), mark a point representing the corresponding \( x \), \( y \), and \( z \) coordinates. Connect these points to get the curve.

Determine the orientation

The orientation of the curve corresponds to the direction in which the parameter \( t \) increases. A way to determine the orientation is to select certain values of \( t \), such as \( t = 0, 1, 2 \), and see how the position moves as \( t \) increases. In this case, as \( t \) increases, \( x \) will increase and \( y \) and \( z \) will also increase.

Key concepts

Curve Sketching

Curve sketching is a crucial aspect of understanding the behavior of mathematical functions, particularly when we want to visualize the path described by a vector-valued function. In our exercise, the sketching process begins with a separation of the vector components associated with the function

\( \bold{r}(t) = t^2\bold{i}+2t\bold{j}+\frac{3}{2}t\bold{k} \). You first need to interpret each component—\(t^2\), \(2t\), and \(\frac{3}{2}t\)—as individual functions of \(t\) in relation to the coordinate axes. When these components are graphed separately, they form their distinct paths: the parabola for the \(x\)-component, and straight lines for the \(y\) and \(z\) components.

Combining these graphs into a three-dimensional sketch involves plotting points for selected \(t\) values and then connecting these points with a smooth curve. This provides a visual representation of the trajectory of the particle, or the path of an object, through space as dictated by the vector function.

Vector Calculus

In vector calculus, vector-valued functions offer a way to model physical quantities that have both magnitude and direction. They are extraordinarily useful in representing trajectories in physics, engineering, and even finance.

The given function
\( \bold{r}(t) = t^2\bold{i}+2t\bold{j}+\frac{3}{2}t\bold{k} \) encapsulates a movement along three dimensions, with each unit vector—\(\bold{i}\), \(\bold{j}\), and \(\bold{k}\)—indicating a direction along the corresponding axis. The coefficients of these vectors are the time-dependent components that provide the exact location in space at any instance \(t\).

Understanding such vector-valued functions allows for the examination of complex motions and understanding the fundamental principles governing them, such as velocity and acceleration, which can be derived by differentiating the position vector with respect to time.

Three-Dimensional Coordinate System

The three-dimensional coordinate system is an extension of the two-dimensional Cartesian plane that most students are familiar with. It's like moving from a flat sheet of paper into the space around us. This system introduces an additional axis—usually labeled as the \(z\)-axis—that extends vertically in relation to the horizontal \(x\)- and \(y\)-axes.

When dealing with a vector-valued function such as
\( \bold{r}(t) = t^2\bold{i}+2t\bold{j}+\frac{3}{2}t\bold{k} \), each component of the function represents a movement along a specific axis. For sketching purposes, a student must consider where a point defined by \((t^2, 2t, \frac{3}{2}t)\) lies within this three-dimensional setup for various values of \(t\).

By plotting enough points and connecting them, you can visualize the curve in space. The three axes intersect at a point called the 'origin,' and any point in space can be defined by its distances along these three perpendicular axes, creating a grid that is the foundation for much of higher mathematics and physics.