91Ó°ÊÓ

In the context of calculus and physics, velocity in a three-dimensional space is a vector quantity that represents the rate of change of a particle's position with respect to time. It combines both the speed and direction of the particle. The position function given here is \( \mathbf{r}(t) = t \mathbf{i} + 2 \cos t \mathbf{j} + \sin t \mathbf{k} \).
To find the velocity vector, which is denoted by \( \mathbf{v}(t) \), you calculate the derivative of the position vector \( \mathbf{r}(t) \) with respect to time \( t \). Through differentiation, we find that: \[ \mathbf{v}(t) = \frac{d}{dt}(t \mathbf{i} + 2 \cos t \mathbf{j} + \sin t \mathbf{k}) = \mathbf{i} - 2 \sin t \mathbf{j} + \cos t \mathbf{k}. \]Here, each component of \( \mathbf{v}(t) \) signifies a directional change:- \( \mathbf{i} \): Change along the x-axis.- \( -2 \sin t \mathbf{j} \): Change along the y-axis.- \( \cos t \mathbf{k} \): Change along the z-axis.These components help in mapping the particle's path over time in 3D space.

Acceleration is another vector quantity and represents the rate of change of velocity with respect to time. It offers insights into how a particle's velocity alters as it traverses its path. To determine the acceleration vector for the given position function, we need to find the derivative of the velocity function \( \mathbf{v}(t) \):\[ \mathbf{a}(t) = \frac{d}{dt}(\mathbf{i} - 2 \sin t \mathbf{j} + \cos t \mathbf{k}) = -2 \cos t \mathbf{j} - \sin t \mathbf{k}. \]The acceleration vector \( \mathbf{a}(t) \) can be broken down as follows:- The \(-2 \cos t \mathbf{j}\) term highlights the radial acceleration component along the y-axis, reflecting periodic slowing and speed-up as influenced by the cosine function.- The \(-\sin t \mathbf{k}\) component shows the influence along the z-axis, changing per the sine function's periodicity.These help in comprehending how the particle's velocity is not steady and changes over the three-dimensional path.

A position vector serves as a detailed description of a point in space. It effectively captures a particle's position in terms of its distance from an origin point in a 3D coordinate system. The given position vector for our exercise is \( \mathbf{r}(t) = t \mathbf{i} + 2 \cos t \mathbf{j} + \sin t \mathbf{k} \).This vector breaks down as:- \( t \mathbf{i} \) describes linear movement along the x-axis, where the particle continues onward as time progresses.- \( 2 \cos t \mathbf{j} \) adds a periodic oscillatory motion along the y-axis, owing to the trigonometric cosine function.- \( \sin t \mathbf{k} \) gives a smooth oscillation up and down along the z-axis, under the sway of the sine function.By analyzing these components, we comprehend how the particle moves via a mixture of constant motion and oscillations, crafting a unique path through 3D space.

Understanding the three-dimensional path of a particle involves piecing together information about its position vector, velocity, and acceleration. The position function \( \mathbf{r}(t) = t \mathbf{i} + 2 \cos t \mathbf{j} + \sin t \mathbf{k} \) helps in outlining this path.Let's break this down:

• The particle moves linearly forward in the direction of the x-axis due to the term \( t \mathbf{i} \).
• It simultaneously oscillates back and forth in the y-axis caused by \( 2 \cos t \mathbf{j} \).
• Additionally, there’s an up and down motion along the z-axis influenced by \( \sin t \mathbf{k} \).

By evaluating these motions at specific points in time (like \(t=0\)), we can sketch a visual representation of the path. This insight is instrumental in illustrating the path the particle crafts through its coordinated linear and oscillatory movements. During \(t=0\), these vectors dictate positioning, velocity, and acceleration, allowing players to graphically map out the particle's trajectory in space.