Question

Find the point P on the curve \(\mathbf{r}(t)\) that lies closest to \(P_{0}\) and state the distance between \(P_{0}\) and \(P\). $$\mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+t \mathbf{k} ; P_{0}(1,1,3)$$

Short answer

Question: Determine the point on the curve with parametric equations \(x = \cos t, y = \sin t, z = t\) that is closest to the point \(P_{0}(1, 1, 3)\), and compute the distance between the two points. Answer: Using the process outlined in the solution, we first need to calculate the critical points of the distance function by setting the first derivative to 0 and solving for t: $$ -2(\cos t - 1)\cdot(-\sin t) + 2(\sin t - 1)\cos t + 2(t - 3) = 0 $$ This equation can be solved using numerical methods or graphically. Suppose we find the value of t using one of these methods. Let's say we found the value of \(t = a\), then we calculate the coordinates of the point P on the curve using this value: $$ P = (\cos a, \sin a, a) $$ Finally, to calculate the distance between \(P_{0}(1, 1, 3)\) and the point P on the curve, we use the distance formula: $$ d = \sqrt{(\cos a - 1)^{2} + (\sin a - 1)^{2} + (a - 3)^{2}} $$

Step-by-step solution

Write the distance function

The distance \(d(t)\) between a point on the curve (\(\mathbf{r}(t)\)) and the point \(P_{0}(1, 1, 3)\) is given by the Euclidean distance formula: $$ d(t) = \sqrt{(\cos t - 1)^{2} + (\sin t - 1)^{2} + (t - 3)^{2}} $$

Simplify the distance function

Instead of working with the square root, which makes the differentiation complicated, we'll work with the square of the distance: $$ d^2(t) = (\cos t - 1)^{2} + (\sin t - 1)^{2} + (t - 3)^{2} $$

Differentiate the simplified distance function

To minimize the distance, we need to find the critical points of the function. To do this, we'll differentiate \(d^2(t)\) with respect to \(t\): $$ \frac{d(d^2(t))}{dt} = -2(\cos t - 1)\cdot(-\sin t) + 2(\sin t - 1)\cos t + 2(t - 3) $$

Set the first derivative to 0 and solve for t

To find the critical points, we set the first derivative equal to 0 and solve for \(t\): $$ -2(\cos t - 1)\cdot(-\sin t) + 2(\sin t - 1)\cos t + 2(t - 3) = 0 $$ Unfortunately, there is no direct algebraic method to solve this equation. However, it can be solved either using numerical methods like the Newton-Raphson method or graphically by analyzing the function.

Find the coordinates of the point P on the curve

Let's assume we found the value of \(t\) from the previous step as \(t=a\). Then, the coordinates of the point P on the curve can be determined by plugging this value into the parametric equations of \(\mathbf{r}(t)\): $$ P = (\cos a, \sin a, a) $$

Calculate the distance between \(P_{0}\) and P

Finally, to calculate the distance between \(P_{0}(1, 1, 3)\) and the point P on the curve, we use the distance formula: $$ d = \sqrt{(\cos a - 1)^{2} + (\sin a - 1)^{2} + (a - 3)^{2}} $$

Key concepts

Euclidean Distance Formula

The Euclidean distance formula is a crucial mathematical tool for finding the shortest distance between two points in a Euclidean space. For two points in a 3-dimensional space, the distance between them can be calculated using the formula:
d(t) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
Where \( (x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) are the coordinates of the two points. It's like measuring the length of a straight line connecting the two points, and it's based on the Pythagorean theorem.
Students often wonder why we square the differences and take the square root in this formula. Squaring ensures that distance is always a non-negative value and taking the square root gives us the actual distance rather than the square of the distance.
In the given exercise, this formula measures the distance between the fixed point \(P_0\) and a point on the curve defined by parametric equations.

Critical Points in Calculus

The term 'critical points' in calculus refers to points on the graph of a function where the derivative is either zero or undefined. These points can indicate where the graph has peaks, troughs, or points of inflection - essentially, they're places where the graph can 'change direction'.
In the context of our problem, identifying the critical points of the squared distance function will help us find where this function reaches a minimum, which corresponds to the shortest distance to the curve.
To find critical points, we first differentiate the function with respect to its variable. Then, we set this derivative equal to zero and solve for the variable. Sometimes, this process can lead to complex equations that aren't easily solvable using standard algebraic methods. When that happens, numerical methods come to the rescue.

Parametric Equations

Parametric equations represent a set of equations that express a group of quantities as explicit functions of a number of independent variables, known as parameters.
In our example, the curve is defined by parametric equations in terms of \(t\), which makes it easier to describe complex curves and motions like circles, ellipses, and even the path of an object in motion.

Why Parametric Equations?

Parametric equations provide flexibility in the representation of curves. While non-parametric equations limit you to functions where each \(x\) has exactly one \(y\), parametric equations have no such constraint. This makes them particularly powerful in physics and engineering, where the position of an object might loop back on itself or involve more complex movement.

Numerical Methods

At times, you'll encounter equations that are impossible or impractical to solve using algebra alone. This is where numerical methods come in - these are techniques used to find approximate solutions to complex mathematical problems.

Real-World Importance

These methods are especially important in the real world, where perfect models rarely exist, and exact solutions are the exception, not the rule.
In the context of our textbook problem, numerical methods like the Newton-Raphson method can be used to find an approximate value of \(t\) that makes the derivative of the squared distance function equal zero, leading us to the minimum distance to the curve. Understanding and applying these methods are essential skills for scientists, engineers, and mathematicians.