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).

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Similar Questions

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.

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