Question

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

Short answer

Calculate derivatives for \( \kappa \), \( \mathbf{T} \), \( \mathbf{N} \), \( \mathbf{B} \) using their formulas.

Step-by-step solution

Find the Derivative of \( \mathbf{r}(t) \)

First, find \( \mathbf{r}'(t) \), the derivative of \( \mathbf{r}(t) \). Since \( \mathbf{r}(t) = 3 \cosh(t/3)\mathbf{i} + t\mathbf{j} \), the derivative is \( \mathbf{r}'(t) = sinh(t/3) \mathbf{i} + \mathbf{j} \).

Evaluate \( \mathbf{r}'(t) \) at \( t = t_1 \)

Substitute \( t = 1 \) into \( \mathbf{r}'(t) \) to get \( \mathbf{r}'(1) = sinh(1/3) \mathbf{i} + \mathbf{j} \).

Find the Magnitude of \( \mathbf{r}'(t) \)

The magnitude \( ||\mathbf{r}'(t)|| = \sqrt{(sinh(t/3))^2 + 1} \), and specifically at \( t = 1 \), it is \( ||\mathbf{r}'(1)|| = \sqrt{(sinh(1/3))^2 + 1} \).

Calculate the Unit Tangent Vector \( \mathbf{T}(t) \)

The unit tangent vector is given by \( \mathbf{T}(t) = \frac{\mathbf{r}'(t)}{||\mathbf{r}'(t)||} \). At \( t = 1 \), it becomes \( \mathbf{T}(1) = \frac{sinh(1/3) \mathbf{i} + \mathbf{j}}{\sqrt{(sinh(1/3))^2 + 1}} \).

Find the Second Derivative \( \mathbf{r}''(t) \)

Find \( \mathbf{r}''(t) \), which is the derivative of \( \mathbf{r}'(t) \). Since \( \mathbf{r}'(t) = sinh(t/3) \mathbf{i} + \mathbf{j} \), the second derivative is \( \mathbf{r}''(t) = (1/3) cosh(t/3) \mathbf{i} \).

Compute the Curvature \( \kappa \)

Curvature \( \kappa \) is calculated using \( \kappa = \frac{||\mathbf{r}'(t) \times \mathbf{r}''(t)||}{||\mathbf{r}'(t)||^3} \). Evaluate this expression at \( t = 1 \).

Calculate \( \mathbf{r}'(t) \times \mathbf{r}''(t) \)

Use the cross product of \( \mathbf{r}'(t) \) and \( \mathbf{r}''(t) \). Since \( \mathbf{r}'(t) = sinh(t/3) \mathbf{i} + \mathbf{j} \) and \( \mathbf{r}''(t) = (1/3)cosh(t/3) \mathbf{i} \), simplify \( \mathbf{r}'(1) \times \mathbf{r}''(1) \).

Find the Unit Normal Vector \( \mathbf{N}(t) \)

The unit normal vector is \( \mathbf{N}(t) = \frac{\mathbf{T}'(t)}{||\mathbf{T}'(t)||} \). Compute the derivative of \( \mathbf{T}(t) \) and evaluate at \( t = 1 \).

Determine the Binormal Vector \( \mathbf{B}(t) \)

The binormal vector \( \mathbf{B}(t) \) is \( \mathbf{T}(t) \times \mathbf{N}(t) \). Calculate this cross product at \( t = 1 \).

Summary

After calculating all the vectors and curvature, summarize the findings for \( \kappa \), \( \mathbf{T}(t) \), \( \mathbf{N}(t) \), and \( \mathbf{B}(t) \) at \( t = 1 \).

Key concepts

Curvature

Curvature is a measure of how much a curve deviates from being a straight line. It tells us how sharply a curve bends at any given point. Mathematically, the curvature, denoted as \( \kappa \), of a curve described by a vector function \( \mathbf{r}(t) \) is given by: \[ \kappa = \frac{||\mathbf{r}'(t) \times \mathbf{r}''(t)||}{||\mathbf{r}'(t)||^3} \] This formula involves taking the cross product of the first derivative \( \mathbf{r}'(t) \) and the second derivative \( \mathbf{r}''(t) \) of the vector function. The norm of this cross product is then divided by the cube of the norm of the first derivative.

• Cross Product: This operation helps in measuring how directional velocities differ along the curve.
• Norm: This gives the length or size of the vector, helping express the magnitude of changes in that direction.

At \( t = 1 \), curvature provides insight into how much the curve \( \mathbf{r}(t) = 3 \cosh(t/3) \mathbf{i} + t \mathbf{j} \) bends, capturing the dynamic shape of the curve effectively.

Unit Tangent Vector

The unit tangent vector, typically represented as \( \mathbf{T}(t) \), is a vector of unit length that points in the direction of the curve's instantaneous velocity. It is essentially the direction of motion along the curve, scaled to have a magnitude, or length, of one. The formula to calculate this vector is: \[ \mathbf{T}(t) = \frac{\mathbf{r}'(t)}{||\mathbf{r}'(t)||} \] To obtain \( \mathbf{T}(t) \), you divide the derivative of \( \mathbf{r}(t) \) by its magnitude.

• Direction: The unit tangent vector indicates the path's forward direction at any given point.
• Magnitude: Normalizing the vector preserves the direction while constraining its length to 1.

At \( t = 1 \), computing \( \mathbf{T}(1) \) instills clarity in understanding the curve's directional tendency at that specific point, helping it align precisely in the velocity's guided path.

Unit Normal Vector

The unit normal vector, denoted by \( \mathbf{N}(t) \), is perpendicular to the unit tangent vector and points in the direction of the curve's principal normal. It illustrates how the curve deviates from its tangent path and aids in comprehending the nature of bending. The unit normal vector is calculated as follows: \[ \mathbf{N}(t) = \frac{\mathbf{T}'(t)}{||\mathbf{T}'(t)||} \] This formula requires differentiating the unit tangent vector \( \mathbf{T}(t) \) and normalizing the result by dividing by its magnitude.

• Perpendicular Relation: The unit normal vector is always orthogonal to the unit tangent vector.
• Bending Direction: Provides insight into the direction in which the curve is turning at a particular point.

By evaluating \( \mathbf{N}(1) \), one can determine how sharply the curve alters course from its tangent direction, highlighting potential curve dynamics at that moment.

Binormal Vector

The binormal vector, symbolized as \( \mathbf{B}(t) \), is crucial in forming the basis for the Frenet-Serret frame together with the tangent and normal vectors. It is perpendicular both to the unit tangent vector and the unit normal vector, providing a complementary directional reference. To find the binormal vector, use the cross product: \[ \mathbf{B}(t) = \mathbf{T}(t) \times \mathbf{N}(t) \] This cross product results in a vector that is orthogonal to both \( \mathbf{T}(t) \) and \( \mathbf{N}(t) \).

• Right-handed System: Constructs a three-dimensional orthogonal system along with \( \mathbf{T}(t) \) and \( \mathbf{N}(t) \).
• Completeness: Finalizes the formation of the local coordinate system describing the space curve.

At \( t = 1 \), calculating \( \mathbf{B}(1) \) provides essential insights into the complete spatial orientation and fluidity of the curve at that particular instant, enhancing spatial analysis understanding.