Find r'(t)*r"(t) given the following vector function. r(t)=(5t^2+5t)i+(5t^2+5t)j

Given the vector function \( \mathbf{r}(t) = (5t^2 + 5t) \hat{i} + (5t^2 + 5t) \hat{j} \), we are tasked with finding \( \mathbf{r}'(t) \cdot \mathbf{r}''(t) \), which is the dot product of the first and second derivatives of the vector function.

### Step 1: Differentiate \( \mathbf{r}(t) \) to find \( \mathbf{r}'(t) \)

The vector function is given as:
\[
\mathbf{r}(t) = (5t^2 + 5t) \hat{i} + (5t^2 + 5t) \hat{j}
\]

Differentiate \( \mathbf{r}(t) \) with respect to \( t \):
\[
\mathbf{r}'(t) = \frac{d}{dt} \left( (5t^2 + 5t) \hat{i} + (5t^2 + 5t) \hat{j} \right)
\]

For the \( \hat{i} \)-component:
\[
\frac{d}{dt}(5t^2 + 5t) = 10t + 5
\]

For the \( \hat{j} \)-component:
\[
\frac{d}{dt}(5t^2 + 5t) = 10t + 5
\]

Thus, we have:
\[
\mathbf{r}'(t) = (10t + 5) \hat{i} + (10t + 5) \hat{j}
\]

### Step 2: Differentiate \( \mathbf{r}'(t) \) to find \( \mathbf{r}''(t) \)

Now, differentiate \( \mathbf{r}'(t) \) with respect to \( t \):
\[
\mathbf{r}''(t) = \frac{d}{dt} \left( (10t + 5) \hat{i} + (10t + 5) \hat{j} \right)
\]

For the \( \hat{i} \)-component:
\[
\frac{d}{dt}(10t + 5) = 10
\]

For the \( \hat{j} \)-component:
\[
\frac{d}{dt}(10t + 5) = 10
\]

Thus, we have:
\[
\mathbf{r}''(t) = 10 \hat{i} + 10 \hat{j}
\]

### Step 3: Compute the dot product \( \mathbf{r}'(t) \cdot \mathbf{r}''(t) \)

The dot product of two vectors \( \mathbf{A} = a_1 \hat{i} + a_2 \hat{j} \) and \( \mathbf{B} = b_1 \hat{i} + b_2 \hat{j} \) is given by:
\[
\mathbf{A} \cdot \mathbf{B} = a_1 b_1 + a_2 b_2
\]

For \( \mathbf{r}'(t) = (10t + 5) \hat{i} + (10t + 5) \hat{j} \) and \( \mathbf{r}''(t) = 10 \hat{i} + 10 \hat{j} \), we have:
\[
\mathbf{r}'(t) \cdot \mathbf{r}''(t) = (10t + 5)(10) + (10t + 5)(10)
\]
\[
\mathbf{r}'(t) \cdot \mathbf{r}''(t) = 100t + 50 + 100t + 50
\]
\[
\mathbf{r}'(t) \cdot \mathbf{r}''(t) = 200t + 100
\]

### Final Answer:
\[
\mathbf{r}'(t) \cdot \mathbf{r}''(t) = 200t + 100
\]

Would you like further details or have any questions?

Here are 5 related questions to explore further:
1. How do you compute the cross product between two vector functions?
2. How would the result change if the vector function had a third component \( \hat{k} \)?
3. What is the physical significance of \( \mathbf{r}'(t) \cdot \mathbf{r}''(t) \) in the context of motion?
4. Can you compute \( \mathbf{r}(t) \times \mathbf{r}'(t) \) for the same vector function?
5. How do you interpret \( \mathbf{r}'(t) \) and \( \mathbf{r}''(t) \) geometrically in terms of motion?

**Tip:** The dot product of two vectors is maximized when the vectors are parallel and minimized when they are perpendicular.

Find r'(t)*r"(t) given the following vector function. r(t) = (5t^2 + 5t)i + (5t^2 + 5t)j

Solve for the dot product of r'(t) and r''(t) for the vector function r(t) = (5t^2 + 5t)i + (5t^2 + 5t)j. This solution uses vector calculus concepts such as derivatives and the dot product, and provides a detailed step-by-step breakdown suitable for undergraduate students.

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