https://file.aiquickdraw.com/mathbot/file/1727454185694pzr900or.jpg

The problem in the image asks for the angle between two vectors given that:

- The scalar (dot) product of the vectors is \( 2 \sqrt{3} \).
- The vector (cross) product of the vectors has a magnitude of \( 2 \).

To solve this, we will use the following relationships:

1. **Dot product formula**: 
   \[
   \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos \theta
   \]
   where \( |\mathbf{A}| \) and \( |\mathbf{B}| \) are the magnitudes of vectors \( \mathbf{A} \) and \( \mathbf{B} \), and \( \theta \) is the angle between them.

2. **Cross product formula**:
   \[
   |\mathbf{A} \times \mathbf{B}| = |\mathbf{A}| |\mathbf{B}| \sin \theta
   \]

Let \( |\mathbf{A}| = a \) and \( |\mathbf{B}| = b \). From the given information:

- \( a b \cos \theta = 2 \sqrt{3} \) (from the scalar product).
- \( a b \sin \theta = 2 \) (from the vector product).

### Step 1: Divide the cross product by the dot product
\[
\frac{a b \sin \theta}{a b \cos \theta} = \frac{2}{2 \sqrt{3}}
\]
This simplifies to:
\[
\tan \theta = \frac{1}{\sqrt{3}}
\]

### Step 2: Solve for \( \theta \)
The angle whose tangent is \( \frac{1}{\sqrt{3}} \) is \( 30^\circ \).

Thus, the angle between the vectors is \( \boxed{30^\circ} \).

Would you like further explanation or more examples on this? Here are 5 related questions to consider:

1. How do the dot and cross products relate geometrically to vectors?
2. What happens when the angle between two vectors is \( 90^\circ \)?
3. How do you find the magnitude of a vector given its components?
4. Can the dot product be zero even if the vectors are not zero vectors?
5. What is the physical interpretation of the cross product in physics?

**Tip:** Remember that the cross product of two vectors gives a vector perpendicular to the plane formed by the two original vectors, with a magnitude proportional to the sine of the angle between them.

If the magnitudes of scalar and vector product of two vectors are 2√3 and 2 respectively, what is the angle between the vectors?

Learn how to calculate the angle between two vectors given the magnitudes of their scalar and vector products. This problem involves the use of dot and cross product formulas, suitable for students in Grades 11-12.

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