A.B=10 A×B=4i+2j-4k tan A dan Tan B adalah

To solve this problem, let's clarify the given information and symbols:

1. **Dot Product (A · B = 10)**:
   \[
   \mathbf{A} \cdot \mathbf{B} = |A| |B| \cos \theta = 10
   \]
   where \( |A| \) and \( |B| \) are the magnitudes of vectors \( \mathbf{A} \) and \( \mathbf{B} \), and \( \theta \) is the angle between them.

2. **Cross Product (A × B = \(4\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}\))**:
   \[
   \mathbf{A} \times \mathbf{B} = |A| |B| \sin \theta \mathbf{n} = 4\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}
   \]
   Here, \( |A| |B| \sin \theta \) represents the magnitude of \( \mathbf{A} \times \mathbf{B} \), and \( \mathbf{n} \) is the unit vector perpendicular to both \( \mathbf{A} \) and \( \mathbf{B} \).

3. **Finding tan A and tan B**:
   Using the dot and cross product, we can find \( \tan \theta \) as follows:
   \[
   \tan \theta = \frac{|\mathbf{A} \times \mathbf{B}|}{\mathbf{A} \cdot \mathbf{B}}
   \]

To proceed, we need to calculate \( |\mathbf{A} \times \mathbf{B}| \) and substitute values into \( \tan \theta \).

### Step 1: Calculate \( |\mathbf{A} \times \mathbf{B}| \)
The magnitude of \( \mathbf{A} \times \mathbf{B} \) is:
\[
|\mathbf{A} \times \mathbf{B}| = \sqrt{4^2 + 2^2 + (-4)^2} = \sqrt{16 + 4 + 16} = \sqrt{36} = 6
\]

### Step 2: Calculate \( \tan \theta \)
Now substitute into \( \tan \theta \):
\[
\tan \theta = \frac{|\mathbf{A} \times \mathbf{B}|}{\mathbf{A} \cdot \mathbf{B}} = \frac{6}{10} = 0.6
\]

So, \( \tan \theta = 0.6 \).

Would you like to see more details or have questions? 

1. How to find the angle \( \theta \) between vectors given dot and cross products?
2. How do magnitudes affect the dot and cross products of vectors?
3. Why is the cross product perpendicular to the original vectors?
4. What are the differences between the dot product and cross product applications?
5. How does the tangent of the angle between vectors relate to their orientation?

**Tip:** Remember, the dot product reveals the angle's cosine (alignment), while the cross product reveals the angle's sine (area or perpendicularity).

A.B=10, A×B=4i+2j-4k. tan A dan Tan B adalah?

Explore how to find the tangent of angles between vectors given their dot product (A · B = 10) and cross product (A × B = 4i + 2j - 4k). Learn key concepts in vector algebra, including dot and cross product formulas and their applications.

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