**Problem Statement:**Find the position function, $\vec{r}(t)$, given that $\vec{r}(0) = \langle 1,1,0 \rangle$, $\vec{v}(0) = \langle 0,0,1 \rangle$ and $\vec{a}(t) = \langle t, e^t, e^{-t} \rangle$.**Explanation:**This problem involves finding the position function $\vec{r}(t)$ from the given initial conditions and the acceleration function. Let's break down the information provided:- $\vec{r}(0) = \langle 1,1,0 \rangle$: This is the initial position vector.- $\vec{v}(0) = \langle 0,0,1 \rangle$: This is the initial velocity vector.- $\vec{a}(t) = \langle t, e^t, e^{-t} \rangle$: This is the acceleration vector as a function of time.To find the position function, we need to integrate the acceleration function twice. First, we find the velocity function $\vec{v}(t)$ by integrating $\vec{a}(t)$ with respect to $t$. Then, we integrate $\vec{v}(t)$ to find the position function $\vec{r}(t)$. Constants of integration will be determined using the initial conditions provided.In more specific steps:1. **Integrate the acceleration function** $\vec{a}(t) = \langle t, e^t, e^{-t} \rangle$ to find the velocity function $\vec{v}(t)$.2. **Determine the constant of integration** for the velocity function using $\vec{v}(0) = \langle 0,0,1 \rangle$.3. **Integrate the resulting velocity function** to find the position function $\vec{r}(t)$.4. **Determine the constant of integration** for the position function using $\vec{r}(0) = \langle 1,1,0 \rangle$.This systematic approach will yield the position vector function $\vec{r}(t)$.This transcription should help users understand the process

We have to find position vector

Answered: < 1,1,0>, v(0) = <0,0,1 > and Find the…

Math

Calculus

< 1,1,0>, v(0) = <0,0,1 > and Find the position function, F(1), given that F(0) = a(t) = < t, e', e¯t >.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

**Problem Statement:**

Find the position function, $\vec{r}(t)$, given that $\vec{r}(0) = \langle 1,1,0 \rangle$, $\vec{v}(0) = \langle 0,0,1 \rangle$ and $\vec{a}(t) = \langle t, e^t, e^{-t} \rangle$.

**Explanation:**

This problem involves finding the position function $\vec{r}(t)$ from the given initial conditions and the acceleration function. Let's break down the information provided:

- $\vec{r}(0) = \langle 1,1,0 \rangle$: This is the initial position vector.
- $\vec{v}(0) = \langle 0,0,1 \rangle$: This is the initial velocity vector.
- $\vec{a}(t) = \langle t, e^t, e^{-t} \rangle$: This is the acceleration vector as a function of time.

To find the position function, we need to integrate the acceleration function twice. First, we find the velocity function $\vec{v}(t)$ by integrating $\vec{a}(t)$ with respect to $t$. Then, we integrate $\vec{v}(t)$ to find the position function $\vec{r}(t)$. Constants of integration will be determined using the initial conditions provided.

In more specific steps:

1. **Integrate the acceleration function** $\vec{a}(t) = \langle t, e^t, e^{-t} \rangle$ to find the velocity function $\vec{v}(t)$.
2. **Determine the constant of integration** for the velocity function using $\vec{v}(0) = \langle 0,0,1 \rangle$.
3. **Integrate the resulting velocity function** to find the position function $\vec{r}(t)$.
4. **Determine the constant of integration** for the position function using $\vec{r}(0) = \langle 1,1,0 \rangle$.

This systematic approach will yield the position vector function $\vec{r}(t)$.

This transcription should help users understand the process

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