1. What is Scalar Triple Product?

The scalar triple product is a mathematical operation that takes three vectors and returns a scalar value. It is defined as the dot product of one vector with the cross product of the other two vectors, and geometrically represents the volume of the parallelepiped formed by the three vectors.

2. How Does the Calculator Work?

The calculator uses the determinant formula:

Which expands to: \[ STP = A_x(B_yC_z - B_zC_y) - A_y(B_xC_z - B_zC_x) + A_z(B_xC_y - B_yC_x) \]

Explanation: The scalar triple product calculates the signed volume of the parallelepiped formed by vectors A, B, and C.

3. Geometric Interpretation

Details: The absolute value of the scalar triple product gives the volume of the parallelepiped formed by the three vectors. If the result is zero, the three vectors are coplanar (lie in the same plane).

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 using the determinant method.

5. Frequently Asked Questions (FAQ)

Q1: What does a positive/negative scalar triple product mean?
A: The sign indicates the orientation of the three vectors. Positive means right-handed system, negative means left-handed system.

Q2: When is the scalar triple product zero?
A: When the three vectors are coplanar (lie in the same plane) or when any two vectors are parallel.

Q3: How is this related to vector triple product?
A: Scalar triple product gives a scalar result, while vector triple product gives a vector result. They are different operations.

Q4: What are practical applications of scalar triple product?
A: Used in physics for torque calculations, in computer graphics for volume calculations, and in engineering for structural analysis.

Q5: Can this be used for 2D vectors?
A: The scalar triple product is specifically defined for 3D vectors. For 2D vectors, different methods are used.