Triple Scalar Product Calculator Symbolab

Triple Scalar Product Formula:

\[ Scalar = \vec{a} \cdot (\vec{b} \times \vec{c}) \]

Vector a (x):

Vector a (y):

Vector a (z):

Vector b (x):

Vector b (y):

Vector b (z):

Vector c (x):

Vector c (y):

Vector c (z):

Triple Scalar Product:

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1. What is the Triple Scalar Product?

The triple scalar product, also known as the scalar triple product, is a mathematical operation that takes three vectors and returns a scalar value. It is calculated as \( \vec{a} \cdot (\vec{b} \times \vec{c}) \) and represents the volume of the parallelepiped formed by the three vectors.

2. How Does the Calculator Work?

The calculator uses the triple scalar product formula:

\[ Scalar = \vec{a} \cdot (\vec{b} \times \vec{c}) \]

Where:

\( \vec{a}, \vec{b}, \vec{c} \) — Three vectors in 3D space
\( \times \) — Cross product operation
\( \cdot \) — Dot product operation

Explanation: First calculate the cross product of vectors b and c, then take the dot product of the result with vector a.

3. Importance of Triple Scalar Product

Details: The triple scalar product is important in vector calculus, physics, and engineering for calculating volumes, determining if vectors are coplanar, and solving various spatial geometry problems.

4. Using the Calculator

Tips: Enter the x, y, and z components of all three vectors. The calculator will compute the cross product of b and c first, then the dot product with a to give the final scalar result.

5. Frequently Asked Questions (FAQ)

Q1: What does the triple scalar product represent geometrically?
A: It represents the signed volume of the parallelepiped formed by the three vectors. A zero result indicates the vectors are coplanar.

Q2: Is the triple scalar product commutative?
A: No, but it has cyclic permutation property: \( \vec{a} \cdot (\vec{b} \times \vec{c}) = \vec{b} \cdot (\vec{c} \times \vec{a}) = \vec{c} \cdot (\vec{a} \times \vec{b}) \).

Q3: What is the significance of a negative result?
A: A negative result indicates the three vectors form a left-handed coordinate system, while positive indicates right-handed.

Q4: Can this be used for 2D vectors?
A: No, the triple scalar product is specifically defined for 3D vectors as it requires three dimensions for meaningful volume calculation.

Q5: How is this related to the determinant?
A: The triple scalar product equals the determinant of the 3×3 matrix formed by the components of the three vectors as rows or columns.