if r=a cos t i+as in t j+t k,then find r=dr/dt;r=d r/dt and

To tackle the problem where the vector \( \mathbf{r} \) is defined as \( \mathbf{r} = a \cos(t) \mathbf{i} + a \sin(t) \mathbf{j} + t \mathbf{k} \), we need to find the derivatives \( \frac{d\mathbf{r}}{dt} \) and \( \frac{d^2\mathbf{r}}{dt^2} \). Let's break this down step by step.

Step 1: Understanding the Components of \( \mathbf{r} \)

The vector \( \mathbf{r} \) consists of three components:

• The \( x \)-component: \( a \cos(t) \)
• The \( y \)-component: \( a \sin(t) \)
• The \( z \)-component: \( t \)

Here, \( a \) is a constant, and \( t \) is the parameter that varies over time.

Step 2: Finding the First Derivative \( \frac{d\mathbf{r}}{dt} \)

To find the first derivative, we differentiate each component of \( \mathbf{r} \) with respect to \( t \):

• For the \( x \)-component:

  \( \frac{d}{dt}(a \cos(t)) = -a \sin(t) \)

• For the \( y \)-component:

  \( \frac{d}{dt}(a \sin(t)) = a \cos(t) \)

• For the \( z \)-component:

  \( \frac{d}{dt}(t) = 1 \)

Putting it all together, we have:

\( \frac{d\mathbf{r}}{dt} = -a \sin(t) \mathbf{i} + a \cos(t) \mathbf{j} + 1 \mathbf{k} \)

Step 3: Calculating the Second Derivative \( \frac{d^2\mathbf{r}}{dt^2} \)

Next, we differentiate \( \frac{d\mathbf{r}}{dt} \) to find the second derivative:

• For the \( x \)-component:

  \( \frac{d}{dt}(-a \sin(t)) = -a \cos(t) \)

• For the \( y \)-component:

  \( \frac{d}{dt}(a \cos(t)) = -a \sin(t) \)

• For the \( z \)-component:

  \( \frac{d}{dt}(1) = 0 \)

Combining these results gives us:

\( \frac{d^2\mathbf{r}}{dt^2} = -a \cos(t) \mathbf{i} - a \sin(t) \mathbf{j} + 0 \mathbf{k} \)

Final Results

In summary, we have derived the following derivatives:

• First derivative:

  \( \frac{d\mathbf{r}}{dt} = -a \sin(t) \mathbf{i} + a \cos(t) \mathbf{j} + 1 \mathbf{k} \)

• Second derivative:

  \( \frac{d^2\mathbf{r}}{dt^2} = -a \cos(t) \mathbf{i} - a \sin(t) \mathbf{j} \)

This process illustrates how to differentiate vector functions with respect to a parameter, which is essential in understanding motion in three-dimensional space. If you have any further questions or need clarification on any part, feel free to ask!