Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the curvature \(\kappa\), the unit tangent vector \(\mathbf{T}\), the unit normal vector \(\mathbf{N}\), and the binormal vector \(\mathbf{B}\) at \(t=t_{1}\). \(\mathbf{r}(t)=3 \cosh (t / 3) \mathbf{i}+t \mathbf{j} ; t_{1}=1\)

Step-by-Step Solution

Answer

Calculate unit vectors \(\mathbf{T}(1)\), \(\mathbf{N}(1)\), and \(\mathbf{B}(1)\), then \(\kappa\) at \(t=1\).

Step 1: Find the velocity vector

To find the velocity vector \(\mathbf{v}(t)\), differentiate the position vector \(\mathbf{r}(t) = 3\cosh\left(\frac{t}{3}\right)\mathbf{i} + t\mathbf{j}\) with respect to \(t\). This yields \(\mathbf{v}(t) = \sinh\left(\frac{t}{3}\right)\mathbf{i} + \mathbf{j}\).

Step 2: Find the acceleration vector

Differentiate the velocity vector \(\mathbf{v}(t) = \sinh\left(\frac{t}{3}\right)\mathbf{i} + \mathbf{j}\) with respect to \(t\) to get the acceleration vector: \(\mathbf{a}(t) = \frac{1}{3}\cosh\left(\frac{t}{3}\right)\mathbf{i}\).

Step 3: Evaluate at \(t=t_1\) for initial velocity

Substitute \(t = 1\) into \(\mathbf{v}(t)\) to get the velocity vector at \(t_1 = 1\): \(\mathbf{v}(1) = \sinh\left(\frac{1}{3}\right)\mathbf{i} + \mathbf{j}\).

Step 4: Calculate the unit tangent vector \(\mathbf{T}\)

The unit tangent vector \(\mathbf{T}(t)\) is \(\frac{\mathbf{v}(t)}{||\mathbf{v}(t)||}\). Find \(||\mathbf{v}(t)||\), the magnitude of \(\mathbf{v}(t)\) at \(t=1\) as \(\sqrt{\sinh^2\left(\frac{1}{3}\right) + 1}\), then evaluate \(\mathbf{T}(1)\).

Step 5: Find the unit normal vector \(\mathbf{N}(t)\)

The unit normal vector \(\mathbf{N}(t)\) is found by differentiating \(\mathbf{T}(t)\), denoted as \(\mathbf{T}'(t)\), and normalizing it, \(\mathbf{N}(t) = \frac{\mathbf{T}'(t)}{||\mathbf{T}'(t)||}\).

Step 6: Calculate the curvature \(\kappa\)

The curvature \(\kappa\) is given by \(\kappa = \frac{||\mathbf{T}'(t)||}{||\mathbf{v}(t)||}\). Compute \(||\mathbf{T}'(t)||\) at \(t=1\) and substitute both values to find \(\kappa\).

Step 7: Determine the binormal vector \(\mathbf{B}(t)\)

The binormal vector \(\mathbf{B}(t)\) is \(\mathbf{T}(t) \times \mathbf{N}(t)\). Use the cross product of \(\mathbf{T}(1)\) and \(\mathbf{N}(1)\) to find \(\mathbf{B}(1)\).

Key Concepts

Unit Tangent Vector

In the world of curves and surfaces, the unit tangent vector, denoted as \( \mathbf{T} \), is essential for understanding the direction in which a curve progresses. This vector is computed by taking the velocity of a curve — which is the derivative of the position vector with respect to time — and normalizing it. This means dividing the vector by its magnitude, which ensures that the tangent vector has a unit length of 1.

• **Calculation:** First, find the velocity vector \( \mathbf{v}(t) \) by differentiating the position vector \( \mathbf{r}(t) \).
• Then, compute its magnitude, \( ||\mathbf{v}(t)|| \), which is the square root of the sum of the squares of its components.
• Finally, the unit tangent vector \( \mathbf{T}(t) \) is given by \( \frac{\mathbf{v}(t)}{||\mathbf{v}(t)||} \).

This vector not only provides direction but also implies how the curve moves in the immediate future from a given point. For example, in the exercise provided, after computing \( ||\mathbf{v}(1)|| \), the unit tangent vector \( \mathbf{T}(1) \) pinpoints the precise direction of the curve when \( t = 1 \).

Unit Normal Vector

The unit normal vector, denoted as \( \mathbf{N} \), provides insight into how a curve bends and twists in space, reflecting the curvature's behavior. After determining the unit tangent vector \( \mathbf{T}(t) \), the next step is to observe how this tangent vector changes. This involves taking its derivative with respect to time, which is \( \mathbf{T}'(t) \).

• **Derivation:** Start by finding \( \mathbf{T}'(t) \), which shows how the tangent vector changes as you move along the curve.
• Afterwards, normalize \( \mathbf{T}'(t) \) to obtain \( \mathbf{N}(t) = \frac{\mathbf{T}'(t)}{||\mathbf{T}'(t)||} \).

The unit normal vector is perpendicular to the unit tangent vector and points towards the curve's center of osculation. It's as if the vector gently nudges from outside the curve, indicating the instantaneous direction of curvature. At \( t = 1 \), you can observe how the unit normal vector \( \mathbf{N}(1) \) specifically characterizes the curve's bend at that moment.

Binormal Vector

The binormal vector, \( \mathbf{B} \), rounds out the trio of the TNB frame (Tangent-Normal-Binormal) which fully describes a curve in three-dimensional space. This vector is derived through the cross product of the unit tangent vector \( \mathbf{T} \) and the unit normal vector \( \mathbf{N} \), resulting in a vector that is perpendicular to both. This perpendicularity ensures \( \mathbf{B} \) always acts at right angles to the plane formed by \( \mathbf{T} \) and \( \mathbf{N} \).

• **Computation:** Find \( \mathbf{B}(t) \) by calculating the cross product \( \mathbf{T}(t) \times \mathbf{N}(t) \).
• The resulting vector \( \mathbf{B}(t) \) is automatically normalized due to the properties of cross products between unit vectors.

The binormal vector provides a final orthogonal direction out of the regular plane of the curve, serving as a fundamental part of defining a positively oriented orthonormal basis for the space surrounding the curve. In our exercise example, calculating \( \mathbf{B}(1) \) gives an understanding of the spatial orientation of the curve at \( t = 1 \).