Question

Give a set of parametric equations that describes the following curves. Graph the curve and indicate the positive orientation. Be sure to specify the interval over which the parameter varies. The piecewise linear path from \(P(-2,3)\) to \(Q(2,-3)\) to \(R(3,5)\)

Short answer

Question: Determine the parametric equations for the path from P(-2,3) to Q(2,-3) to R(3,5), find the positive orientation, and graph the curve. Answer: The piecewise parametric equations for the path from P to Q to R are as follows: For \(0 \leq t_1 \leq 1\): \(x(t_1) = -2 + 4t_1\) \(y(t_1) = 3 - 6t_1\) For \(0 \leq t_2 \leq 1\): \(x(t_2) = 2 + t_2\) \(y(t_2) = -3 + 8t_2\) The positive orientation goes from P to Q to R. To graph the curve, plot the points P(-2,3), Q(2,-3), and R(3,5) on a graph and draw arrows to indicate the positive orientation, with an arrow pointing right and down for PQ, then turning up and slightly to the right for QR.

Step-by-step solution

Find the equation of the line segment PQ

To determine the parametric equations of the line between P(-2,3) and Q(2,-3), we can first find the direction vector PQ: Direction vector PQ = Q - P = (2 - (-2), -3 - 3) = (4, -6) Now, we can write the parametric equations using the initial point P and the direction PQ: \(x(t) = -2 + 4t\) \(y(t) = 3 - 6t\)

Find the equation of the line segment QR

To determine the parametric equations of the line between Q(2,-3) and R(3,5), we can first find the direction vector QR: Direction vector QR = R - Q = (3 - 2, 5 - (-3)) = (1, 8) Now, we can write the parametric equations using the initial point Q and the direction QR: \(x(t) = 2 + t\) \(y(t) = -3 + 8t\)

Specify the parameter interval for each part

Now that we've found the parametric equations for both line segments, let's assign a parameter interval for the path from P to Q and from Q to R. We can set the interval for t from 0 to 1 for each part: for the path between P and Q, we use a separate parameter \(t_1\) in the range \(0 \leq t_1 \leq 1\), and for the path between Q and R, we use a different parameter \(t_2\) in the range \(0 \leq t_2 \leq 1\). So, the piecewise parametric equations will be: For \(0 \leq t_1 \leq 1\): \(x(t_1) = -2 + 4t_1\) \(y(t_1) = 3 - 6t_1\) For \(0 \leq t_2 \leq 1\): \(x(t_2) = 2 + t_2\) \(y(t_2) = -3 + 8t_2\)

Graph the curve and indicate the positive orientation

To graph the curve and show the positive orientation, we can draw the line segments between P(-2,3) and Q(2,-3), and Q(2,-3) and R(3,5) while using arrows to point in the direction indicated by the parametric equations. The positive orientation goes from the starting point P towards Q and then towards R. On a graph, this would be an arrow pointing to the right and down for PQ, then turning up and slightly to the right for QR. So, the path moves from P to Q to R. Please plot the graph and mark the points P(-2,3), Q(2,-3), and R(3,5) on your preferred plotting tool and draw arrows to indicate the positive orientation.

Key concepts

Piecewise Functions

Piecewise functions are mathematical expressions that are defined by different equations or expressions over different intervals. They enable us to describe functions that change behavior across their domains. In the context of parametric equations for curves, piecewise functions become especially helpful for describing paths composed of different segments.

For instance, a path from point \(P(-2,3)\) to \(Q(2,-3)\) and then to \(R(3,5)\) requires different parametric equations for each segment. The piecewise nature comes from transitioning between these segments at specific points or intervals. Here, each segment of the path can be described using its own set of parametric equations, which operate over distinct parameter intervals to clearly outline each path segment.

This method allows us to model more complex paths and shapes by breaking them down into simpler, linear or arcuate sections, and is often applied in computer graphics, robotics, and path planning.

Direction Vectors

Direction vectors are essential tools in describing the orientation and magnitude of line segments in coordinate geometry. They point from one location to another, offering a precise way to define how to traverse from a start point to an endpoint.

To find the direction vector of a line segment connecting two points, such as \(P(-2,3)\) and \(Q(2,-3)\), you subtract the coordinates of the starting point from the coordinates of the endpoint:

• The direction vector from \(P\) to \(Q\) is \((Q_x - P_x, Q_y - P_y)\).
• In this case, \(Q-P = (2 - (-2), -3 - (3)) = (4, -6)\).

Using direction vectors, we can construct parametric equations that start from the initial point and extend in the direction and length specified by the vector. In the context of parametric equations, these vectors help to describe the path made by traversing through sequence of points like \(P\) to \(Q\), and then \(Q\) to \(R\), by characterizing the movement that needs to be made.

Line Segments

In geometry, line segments are the straight parts between two points which are finite in length. They differ from lines that stretch indefinitely but still carry two key characteristics: they have a specific direction and a measurable distance.

When describing line segments using parametric equations, the endpoints and the vector direction are crucial. For example, the line segment from \(P(-2,3)\) to \(Q(2,-3)\) is parametrically described using the start point and its direction vector \((4, -6)\). The equations \(x(t) = -2 + 4t\) and \(y(t) = 3 - 6t\) determine line segment \(PQ\):

• The parameter \(t\) is adjusted from 0 to 1, ensuring that the line segment starts at \(P\) and ends at \(Q\).

Similarly, other segments like \(QR\) can be defined, each with interval parameters, to faithfully model the transition from \(Q\) to \(R\).

By breaking this path into line segments, we simplify the task of modeling and plotting as each segment is a simple linear path. This approach is widely applicable, from mapping courses in navigation systems to drawing graphics in animations.