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To find the dot product (\(A \cdot B\)) of vectors \(\mathbf{A}\) and \(\mathbf{B}\), we use the formula:

\[
\mathbf{A} \cdot \mathbf{B} = (A_x \times B_x) + (A_y \times B_y) + (A_z \times B_z)
\]

Given:

\[
\mathbf{A} = 3\mathbf{i} + 2\mathbf{j} + 4\mathbf{k} \quad \text{and} \quad \mathbf{B} = 2\mathbf{i} - 4\mathbf{j} + 5\mathbf{k}
\]

Let's calculate each component:

1. \(A_x \times B_x = 3 \times 2 = 6\)
2. \(A_y \times B_y = 2 \times (-4) = -8\)
3. \(A_z \times B_z = 4 \times 5 = 20\)

Adding these values together:

\[
\mathbf{A} \cdot \mathbf{B} = 6 + (-8) + 20 = 18
\]

So, the value of \( \mathbf{A} \cdot \mathbf{B} \) is **18**.

Would you like more details or have any questions?

### Related Questions
1. What is the geometric interpretation of the dot product of two vectors?
2. How does the dot product relate to the angle between two vectors?
3. What happens to the dot product if the vectors are perpendicular?
4. How do you compute the cross product of two vectors?
5. How would you find the magnitude of vector \(\mathbf{A}\) or \(\mathbf{B}\)?

### Tip
The dot product is a measure of how much one vector extends in the direction of another; it’s useful for finding angles between vectors and determining orthogonality.

Berapakah nilai A.B dari vektor A = 3i + 2j + 4k, vektor B = 2i - 4j + 5k?

Learn how to calculate the dot product of vectors A = 3i + 2j + 4k and B = 2i - 4j + 5k with a detailed step-by-step solution. This problem involves the dot product formula and vector algebra, suitable for high school students in Grades 10-12.

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