Question

Explain the geometric meaning of \(\mathbf{r}^{\prime}(t)\)

Short answer

Answer: The geometric meaning of \(\mathbf{r}'(t)\), the derivative of a position vector, is that it represents the tangent vector to the path traced by a moving point in space. The length of this tangent vector corresponds to the speed of the particle at time \(t\), and its direction indicates the direction of motion.

Step-by-step solution

Define the position vector

A position vector, denoted as \(\mathbf{r}(t)\), is a function that gives the position of a point in space (either two or three-dimensional space) at a particular time \(t\).

Calculate the derivative of the position vector

To find the derivative of the position vector with respect to time, \(\mathbf{r}'(t)\), we need to find the rate of change of the components of the position vector with respect to time. That is, if \(\mathbf{r}(t) = \langle x(t), y(t), z(t)\rangle\), then \(\mathbf{r}'(t) = \langle x'(t), y'(t), z'(t)\rangle\).

Relate the derivative of the position vector to the tangent vector

The derivative of the position vector, \(\mathbf{r}'(t)\), represents the instantaneous rate of change of the position of a point in space. This rate of change is represented by a vector tangent to the path followed by the point. Specifically, \(\mathbf{r}'(t)\) is the tangent vector to the curve defined by the position vector \(\mathbf{r}(t)\) at the point \(\mathbf{r}(t)\).

Relate the magnitude of the tangent vector to speed

The magnitude of the tangent vector \(\mathbf{r}'(t)\), denoted as \(||\mathbf{r}'(t)||\), is equal to the speed of the particle at time \(t\). This means that the length of the tangent vector represents how fast the particle is moving, while the direction of the tangent vector shows the direction of motion.

Summarize the geometric meaning of \(\mathbf{r}'(t)\)

In conclusion, the geometric meaning of the derivative of the position vector \(\mathbf{r}'(t)\) is that it represents the tangent vector to the path traced by a moving point in space. The length of this tangent vector corresponds to the speed of the particle at time \(t\), and its direction indicates the direction of motion.

Key concepts

Position Vector

A position vector is a fundamental concept in vector calculus, used to describe the location of a point in space. Often denoted by \(\mathbf{r}(t)\), the position vector depends on a parameter, typically time \(t\), which allows us to track the motion of a point as it moves through space.
In a two-dimensional space, a position vector can be written as \(\mathbf{r}(t) = \langle x(t), y(t) \rangle\). This means it has two components, \(x\) and \(y\), which are functions of time. In three-dimensional space, we introduce a third component, \(z(t)\), forming \(\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle\).

• Functionality: The position vector gives the exact location at any given time \(t\).
• Visualization: Imagine it as an arrow stretching from the origin of your coordinate system to the point \((x(t), y(t), z(t))\) at that particular time \(t\).
• Purpose: Enables description of dynamic systems where the spatial point changes over time.

Tangent Vector

The tangent vector provides crucial insight into how a point moves along a curve in vector calculus. When we calculate the derivative of the position vector, represented as \(\mathbf{r}'(t)\), we obtain what is known as a tangent vector. This tangent vector essentially tells us which direction a moving point is heading at any particular moment.
The tangent vector is defined by the rate of change of the position vector's components, that is, \(\mathbf{r}'(t) = \langle x'(t), y'(t), z'(t) \rangle\).

• Instantaneous Velocity: At any time \(t\), the tangent vector indicates the direction and speed of motion.
• Direction of Motion: It points along the path or curve, in the direction the point is moving.
• Relation to Curve: Acts like an arrow that exactly "touches" or is tangent to the curve at a specific point in time.

Understanding the tangent vector allows us to grasp not just the position but also the dynamic nature of moving points in space.

Derivative of a Vector

The derivative of a vector is a key concept in vector calculus, crucial for understanding how vectors change over time. By differentiating the position vector \(\mathbf{r}(t)\) with respect to time, we obtain the tangent vector \(\mathbf{r}'(t)\).
To compute the derivative of the vector \(\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle\), we differentiate each component individually with respect to time: \(\mathbf{r}'(t) = \langle x'(t), y'(t), z'(t) \rangle\).

• Rate of Change: The derivative conveys how fast the components of the vector change with time.
• Directionality: Provides the directional aspect of the motion as it takes into account how the position shifts in all dimensions.
• Application in Physics: In physics, this often corresponds to velocity when the position vector represents a moving object.

Overall, the process of differentiating a vector function is essential for exploring the kinetics of systems, helping to predict and map out paths and motion dynamics effectively.