Question

Give two families of curves in \(\mathbb{R}^{3}\) where \(\mathbf{r}\) and \(\mathbf{r}^{\prime}\) are parallel for all \(t\) in the domain.

Short answer

Answer: The two families of curves in \(\mathbb{R}^3\) with parallel position and velocity vectors are: 1. Straight lines through the origin: \(\mathbf{r}(t) = c\cdot t\), where c is a constant vector. 2. Helices on cylinders oriented along the z-axis: \(\mathbf{r}(t) = \begin{bmatrix} R \cos(\omega t) \\ R \sin(\omega t) \\ c \cdot t \end{bmatrix}\), where R and c are constants and \(\omega\) is the angular velocity.

Step-by-step solution

Family 1: Straight Lines Through the Origin

One simple way to achieve the parallel condition is by considering a straight line that passes through the origin. Let \(\mathbf{r}(t) = c \cdot t\), where c is a constant vector. Then the derivative of position vector with respect to time is: $$\mathbf{r}^\prime(t) = \frac{d}{dt}[c \cdot t] = c$$ Now, we can observe that for this straight line, \(\mathbf{r}(t)\) is always parallel to \(\mathbf{r}^\prime(t)\) since \(\mathbf{r}^\prime(t)\) is a constant multiple of \(\mathbf{r}(t)\). The first family of curves that satisfies the given condition is: $$\mathbf{r}(t) = c\cdot t , \quad \text{where c is a constant vector}$$

Family 2: Helices on Cylinders oriented along the z-axis

Another curve satisfying the parallel condition is a helix on a cylinder with its axis along the z-axis. By parameterizing this helix as follows, we can find that the position and velocity vectors are parallel: $$\mathbf{r}(t) = \begin{bmatrix} R cos(\omega t) \\ R sin(\omega t) \\ c \cdot t \end{bmatrix} , \quad \text{where R and c are constants and}\(\omega\) \text{is the angular velocity}$$ Taking the derivative with respect to time for each component, we get: $$\mathbf{r}^\prime(t) = \begin{bmatrix} -R\omega sin(\omega t) \\ R\omega cos(\omega t) \\ c \end{bmatrix}$$ Now, we can observe that \(\mathbf{r}(t)\) and \(\mathbf{r}^\prime(t)\) are parallel. To show this, let's verify that they are scalar multiples of each other: $$\frac{R\cos(\omega t)}{-R\omega\sin(\omega t)} = \frac{R\sin(\omega t)}{R\omega\cos(\omega t)} = \frac{c\cdot t}{c}$$ Since this relationship holds, the second family of curves that satisfies the given condition is: $$\mathbf{r}(t) = \begin{bmatrix} R \cos(\omega t) \\ R \sin(\omega t) \\ c \cdot t \end{bmatrix} , \quad \text{where R and c are constants and}\(\omega\) \text{is the angular velocity}$$

Key concepts

Parallel vectors

In mathematics, two vectors are parallel if they point in the same direction or directly opposite directions. In other words, two vectors \( \mathbf{a} \) and \( \mathbf{b} \) are parallel if there exists a scalar \( k \) such that \( \mathbf{a} = k \cdot \mathbf{b} \). This implies that their direction is the same up to a positive or negative scaling, although their magnitudes might differ.
A common scenario in \( \mathbb{R}^3 \) involves position vectors and their derivatives, particularly when describing curves. When the derivative of a position vector of a curve, \( \mathbf{r}'(t) \), is parallel to the position vector, \( \mathbf{r}(t) \), it suggests that these vectors share a directional relationship through every point \( t \) in the domain.

• For straight lines, the position vector is always a scalar multiple of the constant direction vector.
• For curves like helices, the changing position vector can still be parallel in terms of maintaining directional consistency through a scalar multiple relationship.

This concept is crucial as it directly links the geometric shape of the curve to its algebraic properties.

Position vector

A position vector is a vector that indicates the position of a point in space relative to an origin. In three-dimensional space, this is typically described by \( \mathbf{r}(t) = \begin{bmatrix} x(t) \ y(t) \ z(t) \end{bmatrix} \), a function of one variable \( t \).
This concept is particularly powerful when considering curves in space. As \( t \) changes, so does the position vector, tracing out a path or curve. The change in this vector as \( t \) varies helps us understand the nature of the curve itself.

• For example, when looking at lines through the origin, \( \mathbf{r}(t) = c \cdot t \) means the line is directly scaled by the constant vector \( c \).
• In the case of helices, the position vector is a combination of circular motion in the XY-plane and linear motion along the Z-axis, represented by \( \mathbf{r}(t) = \begin{bmatrix} R \cos(\omega t) \ R \sin(\omega t) \ c \cdot t \end{bmatrix} \).

Thus, the position vector serves as a fundamental tool in representing and analyzing curves in three-dimensional space.

Derivative calculation

The derivative measures how a function changes as its input changes. For vectors, the derivative \( \mathbf{r}'(t) \) represents the velocity of a moving point as its position vector \( \mathbf{r}(t) \) changes with respect to time.
This calculation is integral to understanding the dynamics of curves:

• For the line \( \mathbf{r}(t) = c \cdot t \), the derivative \( \mathbf{r}'(t) = c \) shows a constant vector, indicating a uniform velocity in a single direction.
• For a helix on a cylinder, checking each component's derivative reveals circular motion in the \( xy \)-plane and constant velocity along the \( z \)-axis:

\[ \mathbf{r}'(t) = \begin{bmatrix} -R\omega \sin(\omega t) \ R\omega \cos(\omega t) \ c \end{bmatrix} \]By calculating derivatives, we can determine how each element of the curve behaves, assisting in the analysis and understanding of the curve's motion. This concept not only applies to mechanical systems but is essential in various fields such as physics and engineering.