Question

Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique. The line that passes through the points (1,1) and \((3,5),\) oriented in the direction of increasing \(x\)

Short answer

Answer: The parametric equations for the line passing through (1,1) and (3,5) are \(x(t) = 1 + 2t\) and \(y(t) = 1 + 4t\), with \(t \in [0,1]\).

Step-by-step solution

Find the direction vector

The line passes through (1,1) and (3,5). In order to find the direction vector of the line, we need to find the difference between the two points. \ Let \(P(1,1)\) and \(Q(3,5)\). The direction vector, \(\vec{v}\) can be found as follows:\ \(\vec{v} = Q - P\)\ \(\vec{v} = (3-1, 5-1)\)\ \(\vec{v} = (2, 4)\)

Write the parametric equation

Now that we have the direction vector \(\vec{v} = (2,4)\), we can use one of the given points (either (1,1) or (3,5)) as the initial point and write the parametric equation in terms of the parameter \(t\). In this example, we will use the point (1,1).\ Let \(r(t) = (x(t), y(t))\) be the position vector. Then the parametric equation can be written as:\ \(r(t) = P + t\vec{v}\)\ \(r(t) = (1,1) + t(2, 4)\)\ Therefore,\ \(x(t) = 1 + 2t\)\ \(y(t) = 1 + 4t\)

Determine the interval for parameter values

As the line extends infinitely in both directions, there is no unique interval for parameter values. However, we can choose an interval that encompasses the given points.\ For point (1,1), let \(t=0\). Then:\ \(x(0) = 1 + 2(0) = 1\)\ \(y(0) = 1 + 4(0) = 1\) For point (3,5), let \(t=1\). Then:\ \(x(1) = 1 + 2(1) = 3\)\ \(y(1) = 1 + 4(1) = 5\) Thus, we can choose the interval \(t \in [0,1]\). The final parametric equations for the line are:\ \(x(t) = 1 + 2t\)\ \(y(t) = 1 + 4t\)\ with \(t \in [0,1]\).

Key concepts

Direction Vector

In geometry and vector mathematics, the direction vector plays a critical role in defining the orientation of a line or vector in space. In the context of parametric equations, the direction vector—often denoted by \( \vec{v} \)—provides the necessary information on which direction a line moves away from a specific point.

Imagine you're on a journey from point A to B; the direction vector tells you exactly how to head from A to B—the 'line' of your journey. In the given problem, we found the direction vector by subtracting the coordinates of two points on our line: \( \vec{v} = Q - P \). This calculation provides us with a vector that points in the direction in which the line progresses, indicating how to travel from point P to Q in the Cartesian plane.

For the points \( P(1,1) \) and \( Q(3,5) \), the direction vector \( \vec{v} \) is computed as \( (3-1, 5-1) = (2,4) \). It suggests that for each step you take in the parameter 't', you move 2 units in the x-direction and 4 units in the y-direction. This concept is incredibly useful as it simplifies the complexity of describing the line's path through a simple vector notation.

Position Vector

The position vector, often represented as \( r(t) \) in parametric forms, is a vector that describes the exact location of a point on a line for any specific value of the parameter 't'. It's like having a GPS reading for every moment of your journey on the line, pinpointing your location at any given instance.

To construct this vector for the line under consideration, we use an initial point—this could be any point on the line, but for simplicity, we often select one of the points provided, such as \( P(1,1) \) in our exercise. By adding the direction vector scaled by the parameter 't' to this initial point, \( r(t) = P + t\vec{v} \), we obtain the position vector that provides the coordinates of any point along the line as it changes with 't'.

In the exercise, the position vector is given as \( r(t) = (1,1) + t(2,4) \), resulting in the parametric equations \( x(t) = 1 + 2t \) and \( y(t) = 1 + 4t \). This beautifully ties together our initial point and the direction vector with the path traveled along the line, allowing us to track the line's trajectory over time.

Parameter Interval

The parameter interval defines the range of values that the parameter 't' can take. This is akin to setting a timeframe for your journey, marking the start and the end. In parametric equations for lines and curves, the interval dictates the segment of the line or curve that we are interested in observing.

In our exercise, while a line is theoretically infinite in both directions, we often want to focus on a specific portion. By determining the interval for 't', we are essentially 'clipping' the infinite line to observe the interesting part between two points. For the line passing through \( P(1,1) \) and \( Q(3,5) \), assuming 't' starts at 0 at point P and increases to 1 while reaching point Q, we choose the interval \( t \in [0,1] \).

Picking this interval ensures that we are looking at just the slice of the journey from P to Q. Outside this interval, the parametric equations would still describe the line, but it would lead to points that lie beyond the segment connecting our initial points. Understanding the parameter interval is essential for visualizing and analyzing specific parts of geometric objects or paths in higher dimensions.