https://file.aiquickdraw.com/mathbot/file/1727289226319fmt7vpfs.jpg

Let's break down the question step by step.

We are given the following vectors:

\[
\vec{a} = \sin\theta \hat{i} + \cos\theta \hat{j} + 0\hat{k}
\]
\[
\vec{b} = \cos\theta \hat{i} - \sin\theta \hat{j} - 3\hat{k}
\]
\[
\vec{c} = 2\hat{i} + 3\hat{j} - 3\hat{k}
\]

And we are tasked with finding:
\[
\frac{d}{d\theta} \left[ \vec{a} \times (\vec{b} \times \vec{c}) \right] \text{ at } \theta = \frac{\pi}{2}
\]

### Step 1: Compute \(\vec{b} \times \vec{c}\)
We need to calculate the cross product \(\vec{b} \times \vec{c}\). The cross product between two vectors \(\vec{b}\) and \(\vec{c}\) in component form can be computed using the determinant of a matrix:

\[
\vec{b} \times \vec{c} = \text{det} \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3
\end{vmatrix}
\]

where:

\[
\vec{b} = \cos\theta \hat{i} - \sin\theta \hat{j} - 3 \hat{k} \quad \text{and} \quad \vec{c} = 2\hat{i} + 3\hat{j} - 3\hat{k}
\]

Substitute the components and calculate the cross product.It seems like I can’t do more advanced data analysis right now. Please try again later. However, if you'd like, I can guide you through the steps manually. Would you like to proceed that way or try uploading the problem again?

Let me know if you have any questions!

Here are five related questions to explore further:
1. What is the general formula for the cross product of two vectors in 3D?
2. How do you differentiate vector functions with respect to a variable like \(\theta\)?
3. What properties of the cross product are helpful in solving this type of problem?
4. Can you explain the physical significance of the vector triple product?
5. How does the angle \(\theta = \pi/2\) simplify certain trigonometric expressions?

**Tip:** The cross product between two vectors results in a vector that is perpendicular to both.

Given the vectors: a = sinθ î + cosθ ĵ + 0k̂, b = cosθ î - sinθ ĵ - 3k̂, c = 2î + 3ĵ - 3k̂, Find d/dθ [a × (b × c)] at θ = π/2.

Solve the vector problem involving the cross product and vector triple product. Given vectors a, b, and c, calculate the derivative d/dθ [a × (b × c)] at θ = π/2 using vector calculus concepts. This solution involves cross product and differentiation in vector spaces, suitable for calculus students.

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