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.

Question

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.

BlackTom AI · Accepted Answer

We start by examining the position function given: x(t) = a + sin(t) b, where a and b are fixed vectors in R^3 and t ∈ R.

- Since a is constant, the motion is entirely governed by the term sin(t) b. The scalar sin(t) takes values in the interval [-1, 1].
- Therefore the particle’s position is x(t) = a + s b with s ∈ [-1, 1]. This describes all points along the straight line through a in the direction of b, but only for scalar parameters between -1 and 1.
- The endpoints of this range correspond to s = -1 and s = 1, giving x(t) = a - b and x(t) = a + b, respectively. Hence the particle moves back and forth along the line segment joining a - b to a + b.

Given these observations, the statement 'The particle oscillates between a−b and a+b.' accurately captures the motion. If one were to consider motion beyond these endpoints, or motion in a direction not aligned with b, that would contradict the form a + sin(t) b, which confines movement to the line spanned by b and centered at a. Since no other options are provided to compare against, this single statement correctly characterizes the trajectory.

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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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