please derive the sin theta = magnitude of the cross product of u and v divided by the magnitude of u multiplied by the magnitude of v

To derive the relationship:

\[
\sin \theta = \frac{|\mathbf{u} \times \mathbf{v}|}{|\mathbf{u}| |\mathbf{v}|}
\]

### Step-by-step derivation:

#### 1. **Cross Product Formula:**

The magnitude of the cross product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\) is given by:

\[
|\mathbf{u} \times \mathbf{v}| = |\mathbf{u}| |\mathbf{v}| \sin \theta
\]

where:
- \(|\mathbf{u} \times \mathbf{v}|\) is the magnitude of the cross product of \(\mathbf{u}\) and \(\mathbf{v}\),
- \(|\mathbf{u}|\) and \(|\mathbf{v}|\) are the magnitudes (or lengths) of vectors \(\mathbf{u}\) and \(\mathbf{v}\),
- \(\theta\) is the angle between the vectors.

This formula comes from the geometric definition of the cross product, which is based on the area of the parallelogram formed by \(\mathbf{u}\) and \(\mathbf{v}\). The sine of the angle between them scales the base and height of that parallelogram.

#### 2. **Solving for \(\sin \theta\):**

From the equation:

\[
|\mathbf{u} \times \mathbf{v}| = |\mathbf{u}| |\mathbf{v}| \sin \theta
\]

We can isolate \(\sin \theta\) by dividing both sides of the equation by \(|\mathbf{u}| |\mathbf{v}|\):

\[
\sin \theta = \frac{|\mathbf{u} \times \mathbf{v}|}{|\mathbf{u}| |\mathbf{v}|}
\]

#### 3. **Conclusion:**

Thus, the sine of the angle \(\theta\) between two vectors \(\mathbf{u}\) and \(\mathbf{v}\) is equal to the magnitude of the cross product of \(\mathbf{u}\) and \(\mathbf{v}\), divided by the product of their magnitudes.

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Would you like any further details on the cross product or the geometric interpretation?

### Related Questions:
1. What is the difference between the dot product and the cross product of two vectors?
2. How does the cross product behave in three-dimensional space versus two-dimensional space?
3. Can you explain the geometric meaning of the magnitude of the cross product?
4. How can you compute the cross product in terms of vector components?
5. How does the angle between two vectors affect the value of their cross product?

#### Tip: 
The cross product is only defined in 3D space, and its result is a vector perpendicular to the plane containing \(\mathbf{u}\) and \(\mathbf{v}\).

Understand how to derive sin(θ) as the magnitude of the cross product of vectors u and v divided by the product of their magnitudes. This step-by-step solution covers key concepts in vector algebra and trigonometry, suitable for Grades 10-12.

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