1. What is the Triple Product?

The scalar triple product of vectors a, b, and c is defined as a · (b × c). It represents the volume of the parallelepiped formed by the three vectors and is zero if the vectors are coplanar.

2. How Does the Calculator Work?

The calculator uses the triple product formula:

Where:

• \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) — Three-dimensional vectors
• \( \times \) — Cross product operation
• \( \cdot \) — Dot product operation

Explanation: First calculates the cross product of b and c, then takes the dot product of that result with vector a.

3. Importance of Triple Product

Details: The triple product is used in physics and engineering to determine volumes, test for coplanarity of vectors, and solve problems in mechanics and electromagnetism.

4. Using the Calculator

Tips: Enter the x, y, z components for each of the three vectors. The calculator will compute the scalar triple product value.

5. Frequently Asked Questions (FAQ)

Q1: What does the triple product represent geometrically?
A: It represents the signed volume of the parallelepiped formed by the three vectors.

Q2: What does a zero triple product mean?
A: A zero result indicates that the three vectors are coplanar (lie in the same plane).

Q3: Is the triple product commutative?
A: No, but cyclic permutations give the same result: a·(b×c) = b·(c×a) = c·(a×b).

Q4: What's the difference between scalar and vector triple product?
A: Scalar triple product gives a scalar value, while vector triple product a × (b × c) gives a vector.

Q5: Where is this used in real-world applications?
A: Used in computer graphics, physics simulations, structural engineering, and electromagnetic field calculations.