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doc_text = '\n'.join([para.text for para in doc.paragraphs]) doc_textThe document contains a set of questions and mathematical problems related to vector calculus, electromagnetism, and other topics. From the text, the second problem appears to be:

"Determine the value of sin given that the angle between the following vectors is as follows:

1. A = 2 - i; B = -3 + 5j - k
2. C = 3i - j; D = 2i + 3j + k

where i, j, k are unit vectors."

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To solve this, we can use the formula for the cosine of the angle between two vectors:

$\cos \theta = \frac{{\mathbf{A} \cdot \mathbf{B}}}{{|\mathbf{A}| |\mathbf{B}|}}$

Then, since you are asked for $\sin \theta$, we use the identity:

$\sin \theta = \sqrt{1 - \cos^2 \theta}$

Let’s calculate it step by step:

Step 1: Calculate the dot product $\mathbf{A} \cdot \mathbf{B}$

For vectors A and B: $A = 2\hat{i} - \hat{j}, \quad B = -3\hat{i} + 5\hat{j} - \hat{k}$ The dot product $A \cdot B$ is: $A \cdot B = (2)(-3) + (-1)(5) + (0)(-1) = -6 - 5 = -11$

Step 2: Find the magnitudes of A and B

The magnitude of vector $A$: $|A| = \sqrt{(2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}$

The magnitude of vector $B$: $|B| = \sqrt{(-3)^2 + (5)^2 + (-1)^2} = \sqrt{9 + 25 + 1} = \sqrt{35}$

Step 3: Calculate $\cos \theta$

$\cos \theta = \frac{{-11}}{{\sqrt{5} \times \sqrt{35}}} = \frac{{-11}}{{\sqrt{175}}}$

Step 4: Calculate $\sin \theta$

Now, using the identity for $\sin \theta$: $\sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(\frac{{-11}}{{\sqrt{175}}}\right)^2 = 1 - \frac{{121}}{{175}} = \frac{{54}}{{175}}$ $\sin \theta = \sqrt{\frac{{54}}{{175}}} = \frac{{\sqrt{54}}}{{\sqrt{175}}} = \frac{{3\sqrt{6}}}{{\sqrt{175}}}$

Thus, the value of $\sin \theta$ is approximately:

$\sin \theta \approx \frac{{3 \times 2.449}}{{13.229}} \approx 0.553$

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Would you like more detailed steps or further clarification on any part of this? Also, here are some related questions that could expand on this topic:

1. What is the relationship between the dot product and the angle between vectors?
2. How do you compute the angle between two vectors using the cross product?
3. How can you use vector identities to simplify expressions in vector calculus?
4. What is the significance of the sine and cosine functions in physics and engineering?
5. How do you interpret the result of $\sin \theta$ in practical applications like electromagnetism?

Tip: The magnitude of a vector is always positive, and the angle between vectors can range from 0 to 180 degrees, so be careful about the signs in your calculations!