Suppose that r°(t) is a vector-valued function. Prove that d (.(ア'xア") = ア.(ア/xア") dt
Suppose that r°(t) is a vector-valued function. Prove that d (.(ア'xア") = ア.(ア/xア") dt
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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college level
topic: vectors proof, proving statement true
![### Problem 2
Suppose that \( \vec{r}(t) \) is a vector-valued function. Prove that
\[ \frac{d}{dt} \left( \vec{r} \cdot (\vec{r}' \times \vec{r}'') \right) = \vec{r} \cdot (\vec{r}' \times \vec{r}''') \]
Tex Expression:
\[ \frac{d}{dt} \left( \vec{r} \cdot (\vec{r}' \times \vec{r}'') \right) = \vec{r} \cdot (\vec{r}' \times \vec{r}''') \]
Explanation:
Given a vector-valued function \( \vec{r}(t) \), we are to prove the above differential relationship involving the dot product and the cross product of the vector and its derivatives.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F133e9a13-17b5-4a02-b6c6-8d87bd9b7e49%2F54fa2f99-96ab-4dd6-8d7b-64ab1eaae073%2Fxpgf3s7_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 2
Suppose that \( \vec{r}(t) \) is a vector-valued function. Prove that
\[ \frac{d}{dt} \left( \vec{r} \cdot (\vec{r}' \times \vec{r}'') \right) = \vec{r} \cdot (\vec{r}' \times \vec{r}''') \]
Tex Expression:
\[ \frac{d}{dt} \left( \vec{r} \cdot (\vec{r}' \times \vec{r}'') \right) = \vec{r} \cdot (\vec{r}' \times \vec{r}''') \]
Explanation:
Given a vector-valued function \( \vec{r}(t) \), we are to prove the above differential relationship involving the dot product and the cross product of the vector and its derivatives.
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