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The problem asks to find two vector operations between two vectors \(\vec{A}\) and \(\vec{B}\), where:

- The magnitude of vector \(\vec{A} = 4\) units,
- The magnitude of vector \(\vec{B} = 8\) units,
- The angle between the two vectors is \(30^\circ\).

You are asked to calculate:
a. The dot product \(\vec{A} \cdot \vec{B}\),  
b. The cross product \(\vec{A} \times \vec{B}\).

### Solution:

#### a. Dot Product (\(\vec{A} \cdot \vec{B}\)):

The dot product between two vectors is given by the formula:
\[
\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\theta)
\]
Where:
- \( |\vec{A}| = 4 \) units,
- \( |\vec{B}| = 8 \) units,
- \( \theta = 30^\circ \).

Substituting the values:
\[
\vec{A} \cdot \vec{B} = 4 \times 8 \times \cos(30^\circ)
\]
We know that \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \), so:
\[
\vec{A} \cdot \vec{B} = 4 \times 8 \times \frac{\sqrt{3}}{2} = 16\sqrt{3}
\]
Thus, the dot product is:
\[
\vec{A} \cdot \vec{B} = 16\sqrt{3} \text{ units}.
\]

#### b. Cross Product (\(\vec{A} \times \vec{B}\)):

The magnitude of the cross product between two vectors is given by:
\[
|\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin(\theta)
\]
Substituting the known values:
\[
|\vec{A} \times \vec{B}| = 4 \times 8 \times \sin(30^\circ)
\]
Since \( \sin(30^\circ) = \frac{1}{2} \):
\[
|\vec{A} \times \vec{B}| = 4 \times 8 \times \frac{1}{2} = 16 \text{ units}.
\]

Thus, the magnitude of the cross product is \( 16 \) units, and the direction follows the right-hand rule.

### Summary of Results:
- a. \(\vec{A} \cdot \vec{B} = 16\sqrt{3} \) units,
- b. \(|\vec{A} \times \vec{B}| = 16\) units.

Would you like further details on any part? 

Here are 5 related questions you can explore:
1. How does the angle between two vectors affect the dot product?
2. What happens to the cross product if the angle between two vectors is \(90^\circ\)?
3. Can the dot product of two vectors be negative? Under what conditions?
4. How do you interpret the direction of the cross product in three-dimensional space?
5. How is the geometric interpretation of the dot product different from that of the cross product?

**Tip:** The dot product gives a scalar (magnitude), while the cross product results in a vector perpendicular to the plane of the two input vectors.

Dua buah vektor memiliki titik pangkal berimpitan. Masing-masing vektor besarnya adalah A = 4 satuan dan B = 8 satuan. Kedua vektor ini membentuk sudut 30°. Tentukan hasil dari: a. A · B b. A × B

Find the dot product and cross product of two vectors A = 4 units and B = 8 units with an angle of 30°. This problem covers vector operations, including key trigonometric concepts, and is suitable for advanced high school students (Grades 11-12).

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