Question

Find parametric equations for the following curves. Include an interval for the parameter values. The line that passes through the points (1,1) and \((3,5),\) oriented in the direction of increasing \(x\)

Short answer

Question: Determine the parametric equations for a line that passes through the given points (1,1) and (3,5) with the direction of increasing x. Answer: The parametric equations for the line are: \(x(t) = 1 + t\) \(y(t) = 1 + 4t\) Interval for t: \(t \in (-\infty, +\infty)\)

Step-by-step solution

Find the slope and the direction vector

To find the slope of the line, we'll calculate the change in y divided by the change in x. Then, the direction vector will be the difference of the coordinates of the two points. Slope (m) = \(\frac{y_2 - y_1}{x_2 - x_1}\) = \(\frac{5 - 1}{3 - 1}\) = \(\frac{4}{2}\) = 2 Direction vector: \((x_2 - x_1, y_2 - y_1) = (3 - 1, 5 - 1) = (2, 4)\)

Write the parametric equation for x

To write the parametric equation for x, use the equation: \(x (t) = x_1 + t (x_2 - x_1)\) Substituting the values of the given points: \(x(t) = 1 + t(2 - 1)\) Simplified: \(x(t)= 1 + t\)

Write the parametric equation for y

To write the parametric equation for y, use the equation: \(y(t) = y_1 + t(y_2 - y_1)\) Substituting the values of the given points: \(y(t) = 1 + t(5 - 1)\) Simplified: \(y(t)= 1 + 4t\)

Choose an interval for parameter t

We need to find an interval for the parameter t. In this case, since the problem is oriented in the direction of increasing x, we do not have any specific restrictions on the interval. We can choose the interval \((-\infty, +\infty)\) as it will cover the entire line. Parametric equations for the given line are: \(x(t) = 1 + t\) \(y(t) = 1 + 4t\) Interval for t: \(t \in (-\infty, +\infty)\)

Key concepts

Direction Vector

A direction vector is a crucial component in determining the path or trajectory of a line in a two-dimensional or three-dimensional space. When given two points, like \( (1,1) \) and \( (3,5) \), you can find the direction vector by subtracting the coordinates of the first point from the second.
\[ \text{Direction vector} = (x_2 - x_1, y_2 - y_1) = (3 - 1, 5 - 1) = (2, 4) \]
This vector represents how much you move in the x direction and y direction as you travel from the first point to the second. In this case, it means that for every increment by 2 in the x direction, you increment by 4 in the y direction. Direction vectors help define the orientation and angle of a line.

Parametric Form

The parametric form is a way of expressing a set of equations that defines a curve. Instead of using a single equation to describe a curve, the parametric form uses one or more parameters. In our example, we used the parameter \( t \) to represent both x and y coordinates.
Basic parametric equations for our line would look like this:
\[ x(t) = 1 + t \]
\[ y(t) = 1 + 4t \]
Here, \( t \) is the parameter, and you can think of it as a representation of time or any variable that describes the position along the line. Choosing different values for \( t \) will give different points on the line, ensuring we cover the full range of the line.

Line Equations

Mathematically, there are different ways to represent line equations, with parametric equations and slope-intercept forms being popular choices. The parametric equation involves parameters that vary to describe a position on the line. Here, the parametric equations are derived from the general line equation.\[ x(t) = x_1 + t(x_2 - x_1) \] and \[ y(t) = y_1 + t(y_2 - y_1) \]
These take into account both starting points (\( x_1, y_1 \)) and directional changes. When you compare it to the standard y = mx + b form, parametric equations provide more flexibility and clarity in dynamic situations, allowing one to express each axis independently.

Mathematical Intervals

Choosing the appropriate interval for your parameter is important as it defines the range of your solutions. For a line defined parametrically, the interval for \( t \) decides which portion of the line is considered. In our case, we have chosen \( t \) to belong to \( (-\infty, +\infty) \), implying that the line extends infinitely in both directions, covering all possible values of x and y.
This choice is logical when no constraints are given, like physical boundaries. However, if you have specific limits, say between two points, then \( t \) would have finite intervals. Understanding intervals helps in focusing on the desired portion of a curve and is fundamental in calculus and real-world applications.