3 Dimensional Product Calculator

Scalar Triple Product:

\[ \text{Product} = \det \begin{bmatrix} a_x & a_y & a_z \\ b_x & b_y & b_z \\ c_x & c_y & c_z \end{bmatrix} \]

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 the Scalar Triple Product?

The scalar triple product of three vectors a, b, and c is the determinant of the 3×3 matrix formed by these vectors. It represents the signed volume of the parallelepiped formed by the three vectors.

2. How Does the Calculator Work?

The calculator computes the determinant of the matrix:

\[ \text{Product} = \det \begin{bmatrix} a_x & a_y & a_z \\ b_x & b_y & b_z \\ c_x & c_y & c_z \end{bmatrix} \]

Which expands to: \[ a_x(b_yc_z - b_zc_y) - a_y(b_xc_z - b_zc_x) + a_z(b_xc_y - b_yc_x) \]

3. Geometric Interpretation

Details: The absolute value of the scalar triple product equals the volume of the parallelepiped formed by the three vectors. A zero value 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 determinant of the resulting 3×3 matrix.

5. Frequently Asked Questions (FAQ)

Q1: What does a positive/negative product mean?
A: The sign indicates the orientation of the vector triplet (right-handed or left-handed).

Q2: How is this related to the cross product?
A: The scalar triple product equals a · (b × c), the dot product of a with the cross product of b and c.

Q3: What units does the result have?
A: The product has units of length cubed (if the vectors represent spatial vectors with length units).

Q4: When is the scalar triple product zero?
A: When the three vectors are coplanar (linearly dependent).

Q5: What applications does this have?
A: Used in physics, engineering, and computer graphics for volume calculations and orientation tests.