Let
𝑎
,
𝑏
∈
𝑅
3
be fixed, distinct vectors, and let
𝑥
:
𝑅
→
𝑅
3
be the function defined by
𝑥
(
𝑡
)
=
𝑎
+
sin
⁡
(
𝑡
)
𝑏
.

Suppose a particle has position
𝑥
(
𝑡
)
at time
𝑡
seconds.

Which of the following best describes the motion of the particle? There is one correct answer.

Question

Let 
𝑎
,
𝑏
∈
𝑅
3
 be fixed, distinct vectors, and let 
𝑥
:
𝑅
→
𝑅
3
 be the function defined by 
𝑥
(
𝑡
)
=
𝑎
+
sin
⁡
(
𝑡
)
𝑏
.

Suppose a particle has position 
𝑥
(
𝑡
)
 at time 
𝑡
 seconds.

Which of the following best describes the motion of the particle? There is one correct answer.

BlackTom AI · Accepted Answer

The problem describes a particle whose position is x(t) = a + sin(t) b, where a and b are fixed vectors in R^3 and t is time.

First, observe the vector sin(t) multiplies b, so the motion is confined to the line that passes through a in the direction of b, since we are adding a fixed vector a to a scalar multiple of b.

Next, consider the range of sin(t). The sine function takes values between -1 and 1 for all real t. Therefore the term sin(t) b ranges from -b to +b. As a result, x(t) ranges from a + (-b) to a + (+b), i.e., from a - b to a + b.

Putting this together, the particle moves back and forth along the line segment centered at a with endpoints a - b and a + b. The motion is periodic with period 2π since sin(t) is periodic, so the particle oscillates between those two endpoints in a repeating cycle.

Now evaluate the provided option: 'The particle oscillates between a − b and a + b.' This precisely matches the deduced motion: the endpoints are a − b and a + b, and the motion is oscillatory between them along the line in the direction of b.

Therefore, the statement correctly describes the particle's motion. The reasoning hinges on the linear combination structure x(t) = a + sin(t) b and the bounded range of sin(t).

21266

查看解析

登录即可查看完整答案

类似问题

Let a,b∈R3 be fixed, distinct vectors, and let x:R→R3 be the function defined by x(t)=a+sin(t)b. Suppose a particle has position x(t) at time t seconds. Which of the following best describes the motion of the particle? There is one correct answer.

In a consumer society, many adults channel creativity into buying things

Economic stress and unpredictable times have resulted in a booming industry for self-help products

People born without creativity never can develop it

A product has a selling price of $20, a contribution margin ratio of 40% and fixed cost of $120,000. To make a profit of $30,000. The number of units that must be sold is: Type the number without $ and a comma. Eg: 20000

Which of the following statement regarding cost is correct:

Under the assumptions used in cost-volume-profit analysis, as the activity increases:

The CVP and break-even analysis are supported by a range of important assumptions. Which of the following is not an assumption of the CVP:

更多留学生实用工具

考试浏览器助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单