You are given the derivative of a vector function r in the component form is dr (e,3e",-2t). You are also given that r(0)= 2i-j+k . dt a) Determine the vector function r (t) in the form r(t)= (x(t), y (t), z(t) An efficient notation for the vector equation of a straight line in 3D(or 2D) is given by ((t) = a + bt where t is any real number, a is the position vector from the origin to a point on the line and b
You are given the derivative of a vector function r in the component form is dr (e,3e",-2t). You are also given that r(0)= 2i-j+k . dt a) Determine the vector function r (t) in the form r(t)= (x(t), y (t), z(t) An efficient notation for the vector equation of a straight line in 3D(or 2D) is given by ((t) = a + bt where t is any real number, a is the position vector from the origin to a point on the line and b
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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![**Vector Calculus Exercise**
You are given the derivative of a vector function \( \mathbf{r} \) in the component form:
\[
\frac{d\mathbf{r}}{dt} = \langle e^{-t}, 3e^{3t}, -2t \rangle
\]
You are also given that \( \mathbf{r}(0) = 2\mathbf{i} - \mathbf{j} + \mathbf{k} \).
**Problem Statements:**
a) Determine the vector function \( \mathbf{r}(t) \) in the form \( \mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle \).
An efficient notation for the vector equation of a straight line in 3D (or 2D) is given by \( \mathbf{l}(t) = \mathbf{a} + t\mathbf{b} \) where \( t \) is any real number. \( \mathbf{a} \) is the position vector from the origin to a point on the line, and \( \mathbf{b} \) is the direction vector.
b) Write the vector equation of the tangent line to the curve \( C \) generated by \( \mathbf{r}(t) \) at the point \( (2, -1, 1) \) using the above form.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa06fb91f-1445-4bd6-b9dd-268642fc0d22%2F5f7c8b39-fbfd-4aaf-a064-dc06f98e3654%2F2ru42g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Vector Calculus Exercise**
You are given the derivative of a vector function \( \mathbf{r} \) in the component form:
\[
\frac{d\mathbf{r}}{dt} = \langle e^{-t}, 3e^{3t}, -2t \rangle
\]
You are also given that \( \mathbf{r}(0) = 2\mathbf{i} - \mathbf{j} + \mathbf{k} \).
**Problem Statements:**
a) Determine the vector function \( \mathbf{r}(t) \) in the form \( \mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle \).
An efficient notation for the vector equation of a straight line in 3D (or 2D) is given by \( \mathbf{l}(t) = \mathbf{a} + t\mathbf{b} \) where \( t \) is any real number. \( \mathbf{a} \) is the position vector from the origin to a point on the line, and \( \mathbf{b} \) is the direction vector.
b) Write the vector equation of the tangent line to the curve \( C \) generated by \( \mathbf{r}(t) \) at the point \( (2, -1, 1) \) using the above form.
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