Question

Consider the line segment joining the points \(P\left(x_{0}, y_{0}, z_{0}\right)\) and \(Q\left(x_{1}, y_{1}, z_{1}\right)\) a. Find a function \(\mathbf{r}(t)\) for the segment \(P Q\) b. Use the are length formula to find the length of \(P Q\). c. Use geometry (distance formula) to verify the result of part (b).

Short answer

Answer: The arc length of the vector function representing a line segment is equal to the geometric distance between the endpoints of the line segment. This is because the arc length of the vector function calculates the length of the path traced by the function, which, in the case of a line segment, is the straight path between the two points, equivalent to the geometric distance between the endpoints.

Step-by-step solution

Define the vector function for the line segment PQ

We will first find a function \(\mathbf{r}(t)\) for the segment \(PQ\) by creating a vector function. Let \(\mathbf{r}_{P} = \left\langle x_{0}, y_{0}, z_{0}\right\rangle\) be the position vector of point \(P\), and let \(\mathbf{r}_{Q} = \left\langle x_{1}, y_{1}, z_{1}\right\rangle\) be the position vector of point \(Q\). Create a new vector function using the given endpoints \(\mathbf{r}_{P}\) and \(\mathbf{r}_{Q}\), with our variable \(t \in [0, 1]\): $$\mathbf{r}(t) = (1-t)\mathbf{r}_{P} + t\mathbf{r}_{Q} = (1-t)\left\langle x_{0}, y_{0}, z_{0}\right\rangle + t\left\langle x_{1}, y_{1}, z_{1}\right\rangle.$$ Now we can write the explicit vector function \(\mathbf{r}(t)\): $$\mathbf{r}(t) = \left\langle (1-t)x_{0} + tx_{1}, (1-t)y_{0} + ty_{1}, (1-t)z_{0} + tz_{1}\right\rangle.$$

Compute the arc length of PQ

To find the arc length of the vector function \(\mathbf{r}(t)\) representing the line segment \(PQ\), we'll use the arc length formula: $$L(\mathbf{r}) = \int_{0}^{1}\|\mathbf{r}'(t)\| dt,$$ where \(\mathbf{r}'(t)\) is the derivative of \(\mathbf{r}(t)\) with respect to \(t\). First, find the derivative of \(\mathbf{r}(t)\): $$\mathbf{r}'(t) = \frac{d}{dt}\left(\left\langle (1-t)x_{0} + tx_{1}, (1-t)y_{0} + ty_{1}, (1-t)z_{0} + tz_{1}\right\rangle\right) = \left\langle x_{1}-x_{0}, y_{1}-y_{0}, z_{1}-z_{0}\right\rangle.$$ Next, find the magnitude of \(\mathbf{r}'(t)\): $$\|\mathbf{r}'(t)\| = \sqrt{(x_{1} - x_{0})^{2} + (y_{1} - y_{0})^{2} + (z_{1} - z_{0})^{2}}.$$ Now, integrate this over \(t\) from \(0\) to \(1\) to find the arc length of \(PQ\): $$L(\mathbf{r}) = \int_{0}^{1}\sqrt{(x_{1} - x_{0})^{2} + (y_{1} - y_{0})^{2} + (z_{1} - z_{0})^{2}} dt.$$ Since the integrand is constant, we can write the length \(L(\mathbf{r})\) as: $$L(\mathbf{r}) = \sqrt{(x_{1} - x_{0})^{2} + (y_{1} - y_{0})^{2} + (z_{1} - z_{0})^{2}}.$$

Verify the length using the geometric distance formula

Now we will verify that the length of the line segment \(PQ\) is the same using the geometric distance formula: $$D(P, Q) = \sqrt{(x_{1} - x_{0})^{2} + (y_{1} - y_{0})^{2} + (z_{1} - z_{0})^{2}}.$$ Comparing this to our result from Step 2, we can see that the two results are the same: $$D(P, Q) = L(\mathbf{r}).$$ So, the arc length of the vector function \(\mathbf{r}(t)\) representing the line segment \(PQ\) is equal to the geometric distance between points \(P\) and \(Q\).

Key concepts

Arc Length Formula

The arc length formula is a fundamental concept in calculus, particularly in vector calculus. Its purpose is to find the length of a curve or line segment that cannot be measured directly using standard geometry tools.

When dealing with a vector function \(\mathbf{r}(t)\), which describes a curve in space, the arc length from \(t = a\) to \(t = b\) is given by:
\[L(\mathbf{r}) = \int_{a}^{b}\|\mathbf{r}'(t)\| dt,\]
where \(\mathbf{r}'(t)\) is the derivative of the vector function \(\mathbf{r}(t)\) with respect to \(t\), and \(\|\mathbf{r}'(t)\|\) is the magnitude (or Euclidean norm) of the derivative vector. Intuitively, this formula sums up infinitesimal line segments along the curve to determine its total length, similar to how you would measure the length of a bent wire. This is especially useful for more complex curves where direct measurement is either difficult or impossible.

Line Segment Calculation

A line segment calculation is typically much simpler than curved line calculations because the path between two points is straight. To represent a line segment in vector form, which is the path from point \(P\) to point \(Q\), we use the parametric equation:
\[\mathbf{r}(t) = (1-t)\mathbf{r}_{P} + t\mathbf{r}_{Q},\]
where \(t\) is a parameter varying between 0 and 1, \(\mathbf{r}_{P}\) and \(\mathbf{r}_{Q}\) are position vectors for points \(P\) and \(Q\) respectively. In the specific exercise, the vector function simplifies the process of finding the length of the segment by reducing it to evaluating the integral of a constant.

Vector Calculus

Vector calculus extends the concepts of calculus to vector fields and vector functions. It is essential in many branches of physics and engineering, as it provides the tools to study motion and change in multidimensional systems. Vector functions like \(\mathbf{r}(t)\) describe not only the path of an object but potentially its speed and acceleration as functions of time or another variable.

In the context of line segments and arc lengths, vector calculus simplifies complex spatial problems into manageable calculations. It allows for parametrization of paths, which leads to easier manipulation and understanding of the curve or line's properties. By using derivatives and integrals of vector functions, we can extract valuable information about the curves, such as their length, as demonstrated in the exercise.

Geometric Distance Formula

The geometric distance formula provides a straightforward method to determine the distance between two points in Euclidean space. It's derived from the Pythagorean theorem and is essential for calculations involving straight lines. The formula for the distance between two points \(P\) and \(Q\) with coordinates \(P(x_{0}, y_{0}, z_{0})\) and \(Q(x_{1}, y_{1}, z_{1})\) respectively is:
\[D(P, Q) = \sqrt{(x_{1} - x_{0})^{2} + (y_{1} - y_{0})^{2} + (z_{1} - z_{0})^{2}}.\]
This formula was indeed used to verify the result obtained from the arc length calculation of the vector function representing the line segment in the exercise. This connection between vector functions and geometry is a beautiful example of how various mathematical concepts can complement and confirm each other.