1. What is the Scalar Triple Product?

The scalar triple product of three vectors A, B, and C is defined as A · (B × C). It represents the signed volume of the parallelepiped formed by the three vectors and is a scalar quantity.

2. How Does the Calculator Work?

The calculator uses the scalar triple product formula:

Where:

• \( \mathbf{A}, \mathbf{B}, \mathbf{C} \) — Three vectors in 3D space
• \( \times \) — Cross product operation
• \( \cdot \) — Dot product operation

Calculation Steps:

1. Calculate the cross product of vectors B and C
2. Calculate the dot product of vector A with the result from step 1
3. The result is a scalar value representing the volume

3. Applications of Scalar Triple Product

Details: The scalar triple product has important applications in physics and engineering, including calculating volumes, determining if vectors are coplanar, and solving problems in mechanics and electromagnetism.

4. Using the Calculator

Tips: Enter the x, y, and z components for each of the three vectors. The calculator will compute the scalar triple product using the formula A · (B × C).

5. Frequently Asked Questions (FAQ)

Q1: What does a zero scalar triple product indicate?
A: A zero result indicates that the three vectors are coplanar (lie in the same plane).

Q2: Is the scalar triple product commutative?
A: No, the operation is not commutative. A · (B × C) = B · (C × A) = C · (A × B), but changing the order cyclically changes the sign.

Q3: What is the geometric interpretation?
A: The absolute value of the scalar triple product equals the volume of the parallelepiped formed by the three vectors.

Q4: Can this be used with 2D vectors?
A: No, the scalar triple product is specifically defined for three vectors in 3D space.

Q5: How is this related to the determinant?
A: The scalar triple product equals the determinant of the 3×3 matrix whose rows (or columns) are the components of the three vectors.