Scalar Triple Vector Calculator

Scalar Triple Product Formula:

\[ STP = \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \]

Vector a (x,y,z):

Vector b (x,y,z):

Vector c (x,y,z):

Scalar Triple Product:

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1. What is a Scalar Triple Product?

Definition: The scalar triple product of three vectors a, b, and c is defined as a · (b × c), which gives a scalar quantity.

Purpose: It calculates the volume of the parallelepiped formed by the three vectors and tests for vector coplanarity.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ STP = \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \]

Where:

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

Explanation: First computes the cross product of b and c, then takes the dot product of a with this result.

3. Importance of Scalar Triple Product

Details: The scalar triple product gives the signed volume of the parallelepiped formed by the three vectors. A value of zero indicates the vectors are coplanar.

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.

5. Frequently Asked Questions (FAQ)

Q1: What does the sign of the result mean?
A: The sign indicates the orientation of the three vectors (right-handed or left-handed system).

Q2: What does a zero result indicate?
A: A zero result means the three vectors are coplanar (lie in the same plane).

Q3: How is this different from vector triple product?
A: The scalar triple product returns a scalar value, while the vector triple product a × (b × c) returns a vector.

Q4: Can I use 2D vectors?
A: No, all three vectors must be 3-dimensional for this calculation.

Q5: What units does the result have?
A: The units are the product of the units of all three vector components (e.g., m³ if inputs are in meters).