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Let the position vector (with its tail at the origin) of a moving particle be r = r(t) = t²i - 2tj + (t² + 2t)k, where t represents time. (a) Show that the particle goes through the point (4, -4,8). At what time does it do this? (b) (c) Find the equations of the line tangent to the curve described by the particle and the plane normal to this curve, at the point (4, -4,8). Find the velocity vector and the speed of the particle at time t; at the time when it passes though the point (4, -4,8).

Let the position vector (with its tail at the origin) of a moving particle be r = r(t) = t²i - 2tj + (t² + 2t)k, where t represents time. (a) Show that the particle goes through the point (4, -4,8). At what time does it do this? (b) (c) Find the equations of the line tangent to the curve described by the particle and the plane normal to this curve, at the point (4, -4,8). Find the velocity vector and the speed of the particle at time t; at the time when it passes though the point (4, -4,8).

Calculus: Early Transcendentals
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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**Problem: Position and Motion of a Particle** Consider a moving particle whose position vector (with its tail at the origin) is given by \( \mathbf{r} = \mathbf{r}(t) = t^2 \mathbf{i} - 2t \mathbf{j} + (t^2 + 2t) \mathbf{k} \), where \( t \) represents time. **Tasks:** **(a)** Show that the particle passes through the point \( (4, -4, 8) \). Determine the time at which this occurs. **(b)** Calculate the velocity vector and the speed of the particle at time \( t \), specifically when it passes through the point \( (4, -4, 8) \). **(c)** Find the equations for both the line tangential to the curve described by the particle and the plane normal to this curve at the point \( (4, -4, 8) \).
Transcribed Image Text:**Problem: Position and Motion of a Particle** Consider a moving particle whose position vector (with its tail at the origin) is given by \( \mathbf{r} = \mathbf{r}(t) = t^2 \mathbf{i} - 2t \mathbf{j} + (t^2 + 2t) \mathbf{k} \), where \( t \) represents time. **Tasks:** **(a)** Show that the particle passes through the point \( (4, -4, 8) \). Determine the time at which this occurs. **(b)** Calculate the velocity vector and the speed of the particle at time \( t \), specifically when it passes through the point \( (4, -4, 8) \). **(c)** Find the equations for both the line tangential to the curve described by the particle and the plane normal to this curve at the point \( (4, -4, 8) \).
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