Prove that the curve r(t) = (a + bt",c + dt",e+ ft), where a, b, c, d, e, and f are real numbers and p is a positive integer, has zero curvature. Give an explanation. What must be shown to prove that r(t) has zero curvature? O A. It must be shown that the velocity, v, and the acceleration, a, are constant. GB. It must be shown that the magnitude of the cross product, Ja xv, is zero, or that the unit tangent vector, T, is constant. OC. It must be shown that the cross product a xv is constant. O D. It must be shown that the dot product, a• v, is zero, or that the acceleration, a, is constant. In this problem, show that the magnitude of the cross product, axv, is zero. To do so, first find the velocity, v. r = {a+bt°,c+ dt,e+
Prove that the curve r(t) = (a + bt",c + dt",e+ ft), where a, b, c, d, e, and f are real numbers and p is a positive integer, has zero curvature. Give an explanation. What must be shown to prove that r(t) has zero curvature? O A. It must be shown that the velocity, v, and the acceleration, a, are constant. GB. It must be shown that the magnitude of the cross product, Ja xv, is zero, or that the unit tangent vector, T, is constant. OC. It must be shown that the cross product a xv is constant. O D. It must be shown that the dot product, a• v, is zero, or that the acceleration, a, is constant. In this problem, show that the magnitude of the cross product, axv, is zero. To do so, first find the velocity, v. r = {a+bt°,c+ dt,e+
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Prove that the curve** \( \mathbf{r}(t) = \langle a + bt^p, c + dt^p, e + ft^p \rangle \), **where a, b, c, d, e, and f are real numbers and p is a positive integer, has zero curvature. Give an explanation.**
**What must be shown to prove that \(\mathbf{r}(t)\) has zero curvature?**
- **A.** It must be shown that the velocity, \(\mathbf{v}\), and the acceleration, \(\mathbf{a}\), are constant.
- **B.** It must be shown that the magnitude of the cross product, \(|\mathbf{a} \times \mathbf{v}|\), is zero, or that the unit tangent vector, \(\mathbf{T}\), is constant.
- **C.** It must be shown that the cross product \(\mathbf{a} \times \mathbf{v}\) is constant.
- **D.** It must be shown that the dot product, \(\mathbf{a} \cdot \mathbf{v}\), is zero, or that the acceleration, \(\mathbf{a}\), is constant.
**In this problem, show that the magnitude of the cross product, \(|\mathbf{a} \times \mathbf{v}|\), is zero. To do so, first find the velocity, \(\mathbf{v}\).**
\[ \mathbf{r} = \langle a + bt^p, c + dt^p, e + ft^p \rangle \]
\[ \mathbf{v} = \langle \ \ \ \ \ \ \ \ \ \rangle \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fca46498c-9039-4635-914d-0b10c3becf43%2Fd428353d-3858-4481-ac1f-21b686830480%2Ftt6hdx_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Prove that the curve** \( \mathbf{r}(t) = \langle a + bt^p, c + dt^p, e + ft^p \rangle \), **where a, b, c, d, e, and f are real numbers and p is a positive integer, has zero curvature. Give an explanation.**
**What must be shown to prove that \(\mathbf{r}(t)\) has zero curvature?**
- **A.** It must be shown that the velocity, \(\mathbf{v}\), and the acceleration, \(\mathbf{a}\), are constant.
- **B.** It must be shown that the magnitude of the cross product, \(|\mathbf{a} \times \mathbf{v}|\), is zero, or that the unit tangent vector, \(\mathbf{T}\), is constant.
- **C.** It must be shown that the cross product \(\mathbf{a} \times \mathbf{v}\) is constant.
- **D.** It must be shown that the dot product, \(\mathbf{a} \cdot \mathbf{v}\), is zero, or that the acceleration, \(\mathbf{a}\), is constant.
**In this problem, show that the magnitude of the cross product, \(|\mathbf{a} \times \mathbf{v}|\), is zero. To do so, first find the velocity, \(\mathbf{v}\).**
\[ \mathbf{r} = \langle a + bt^p, c + dt^p, e + ft^p \rangle \]
\[ \mathbf{v} = \langle \ \ \ \ \ \ \ \ \ \rangle \]
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