Chapter 11: Q. 61 (page 873)

Let $r (t)$ be a vector-valued function whose graph is a curve C, and let $a (t)$ be the acceleration vector. Prove that if $a (t)$ is always zero, then C is a straight line.

Short Answer

Expert verified

Ans: If $a (t) = ⟨ 0, 0, 0 ⟩$ then $r (t) = <c_{1} t α, c_{2} t β, c_{3} t + γ>$ the graph is a straight line.

Step by step solution

Step 1. Given information:

The graph of the vectors valued function $r (t)$ is on curve C. $a (t)$ be the acceleration vector.

Step 2. Proving C is a straight line:

Let $a (t) = ⟨ 0, 0, 0 ⟩$

The velocity vector, $v (t) = \int a (t) d t$

$= \int ⟨ 0, 0, 0 ⟩ d t$

$= <c_{1}, c_{2}, c_{3}> where c_{1}, c_{2} and c_{3} are constants r (t) = \int v (t) d t = \int <c_{1}, c_{2}, c_{3}> d t = \int c_{1} d t i + \int c_{2} d t j \int c_{3} d t k$

$= (c_{1} t + α) i + (c_{2} t + β) j + (c_{3} t + γ) k$ where $α, β, γ$ are constants.

$r (t) = <c_{1} t α, c_{2} t β, c_{3} t + γ>$ , the graph is a straight line because it is in linear form.

Thus if the acceleration vector of $r (t)$ is always zero, then the graph of $r (t), c$ , is a straight line.

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Let $r (t)$ be a vector-valued function whose graph is a curve C, and let $a (t)$ be the acceleration vector. Prove that if $a (t)$ is always zero, then C is a straight line.

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Proving C is a straight line:

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Let r(t)be a vector-valued function whose graph is a curve C, and let a(t)be the acceleration vector. Prove that if a(t)is always zero, then C is a straight line.

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Proving C is a straight line:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

Let $r (t)$ be a vector-valued function whose graph is a curve C, and let $a (t)$ be the acceleration vector. Prove that if $a (t)$ is always zero, then C is a straight line.