Triple Vector Cross Product Calculator Symbolab

Triple Vector Product Formula:

\[ a \cdot (b \times c) = a_x (b_y c_z - b_z c_y) - a_y (b_x c_z - b_z c_x) + a_z (b_x c_y - b_y c_x) \]

Vector a:

X component:

Y component:

Z component:

Vector b:

X component:

Y component:

Z component:

Vector c:

X component:

Y component:

Z component:

Triple Product Result:

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1. What is the Triple Vector Product?

The triple vector product, also known as the scalar triple product, calculates the volume of the parallelepiped formed by three vectors. It's defined as a · (b × c) and results in a scalar value.

2. How Does the Calculator Work?

The calculator uses the triple product formula:

\[ a \cdot (b \times c) = a_x (b_y c_z - b_z c_y) - a_y (b_x c_z - b_z c_x) + a_z (b_x c_y - b_y c_x) \]

Where:

a, b, c — Three vectors in 3D space
× — Cross product operator
· — Dot product operator
The result is a scalar value representing volume

3. Applications of Triple Product

Details: The triple product is used in physics, engineering, and computer graphics for calculating volumes, determining if vectors are coplanar, and solving various vector equations.

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 a · (b × c).

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 result mean?
A: A zero result indicates the three vectors are coplanar (lie in the same plane).

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

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

Q5: How is this used in physics?
A: It's used in calculating torque, angular momentum, and in electromagnetic theory.