Question

Sketch the curve represented by the vectorvalued function and give the orientation of the curve. \(\mathbf{r}(t)=(-t+1) \mathbf{i}+(4 t+2) \mathbf{j}+(2 t+3) \mathbf{k}\)

Short answer

The curve will show an orientation from the region of space where x values are high, towards where the y and z values are high. It would move roughly from the (x) direction to the (y, z) plane as t increases. For precise results, a 3D plotter or software should be used.

Step-by-step solution

Identify the Components of the Vector

The vector-valued function is \( \mathbf{r}(t)=(-t+1)\mathbf{i}+(4t+2)\mathbf{j}+(2t+3)\mathbf{k} \). Break this function into its i, j, and k components: x(t) = -t + 1, y(t) = 4t + 2, and z(t) = 2t + 3.

Plot the Components

Since it's a 3D vector, plotting might not be straightforward, you need a 3D plotter for representation. The function x(t) = -t + 1 is a straight line that decreases as t increases. The function y(t) = 4t + 2 is a line that represents a positive slope and increases as t increases. Lastly, z(t) = 2t + 3 also represents a positive slope and so, increases as t increases.

Find the Orientation

Orientation will depend on the direction of increasing parameter t, which is from the point where t=0 to where t increases. It can be seen that as t increases, x decreases, y and z increase. This determines the orientation of the curve in the 3D space.

Key concepts

Sketching Curves in Calculus

When dealing with calculus and vector-valued functions, sketching curves is a vital skill for visualizing and understanding the behavior of functions in space. To begin with, let's recall that a vector-valued function like \( \mathbf{r}(t)=(-t+1) \mathbf{i}+(4 t+2) \mathbf{j}+(2 t+3) \mathbf{k} \) can represent a curve in three-dimensional space. Each component of the function corresponds to one dimension: the \( \mathbf{i} \) component is the x-coordinate, the \( \mathbf{j} \) component is the y-coordinate, and the \( \mathbf{k} \) component is the z-coordinate.

To sketch such a curve, one systematic approach is to evaluate the function at various points of \( t \) to plot corresponding points in 3D space. By connecting these points, you begin to see the trajectory of the curve. It's also beneficial to determine key information, like where the function crosses the coordinate axes, which occurs when the other two variables are zero. Additionally, you can determine the behavior of the curve by examining the limits as \( t \) approaches infinity or negative infinity. Understanding this can provide insight into whether the curve extends infinitely in a particular direction or loops back onto itself.

For the given vector-valued function, we identify the individual components \( x(t) \) , \( y(t) \) , and \( z(t) \) to get a sense of how the curve behaves with respect to each axis separately. This step-by-step analysis allows for a more comprehensive sketching of the curve, considering its full dynamic nature in 3D space.

3D Graph Plotting

Graphing in three dimensions adds a layer of complexity to understanding mathematical functions compared to two-dimensional graphing. A 3D plot allows us to see the complete form of a vector-valued function, which is often impossible to fully grasp with a flat, two-dimensional representation.

To plot the given vector-valued function \( \mathbf{r}(t) \) in 3D, you would typically use software tools or graphing calculators that can handle three-dimensional graphing. By inputting the separate component functions ( \( x(t) \) , \( y(t) \) , and \( z(t) \) ), the tool can render the curve in a space defined by x, y, and z axes.

When plotted, functions like \( x(t) = -t + 1 \) will generally represent a plane or a line within the 3D space. In our case, all three component functions represent lines. Therefore, when these lines are plotted concerning the parameter \( t \) , they together form a curve in space. The beauty of 3D graph plotting is that we can rotate the graph, zoom in and out, and dissect the function in various planes to understand the full geometry of the curve being represented.

For educational purposes, it is beneficial to discuss the use of such tools and encourage students to explore graphs from multiple angles to better internalize the concept of multi-dimensional shapes and their representations.

Vector Orientation

The concept of vector orientation refers to the direction in which the vector points, which is particularly important when representing curves in space. In the context of a vector-valued function, the orientation of the curve is determined by the direction in which the parameter \( t \) increases.

For our function \( \mathbf{r}(t) \) , as \( t \) increases, the x-values \( -t+1 \) are decreasing, whereas the y-values \( 4t+2 \) and z-values \( 2t+3 \) are increasing. This tells us that as we move along the curve in the direction of increasing \( t \) , we are moving in a direction where the curve is descending along the x-axis and ascending along the y and z-axes.

The orientation can have significant consequences in applications such as physics, where it may represent a direction of force or velocity. In calculus, understanding the orientation is crucial when computing line integrals, as the positive direction of the curve will affect the result of the integral.

Visualizing or indicating the orientation on a sketch can be done by drawing arrows along the curve. If you were to sketch the curve on paper, using arrows to indicate the direction where \( t \) increases, provides a clear visual cue of the curve's orientation. This practice enhances comprehension and assists in solving more complex problems where directionality becomes a fundamental aspect of the solution.