Triple Vector Cross Product Calculator Wolfram Alpha

Triple Vector Cross Product Formula:

\[ a \cdot (b \times c) = \det(\begin{bmatrix} a \\ b \\ c \end{bmatrix}) \]

Vector a (x component):

Vector a (y component):

Vector a (z component):

Vector b (x component):

Vector b (y component):

Vector b (z component):

Vector c (x component):

Vector c (y component):

Vector c (z component):

Triple Product (Scalar):

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1. What is the Triple Vector Cross Product?

The triple vector cross product, also known as the scalar triple product, calculates the volume of the parallelepiped formed by three vectors. It's given by \( a \cdot (b \times c) \) and equals the determinant of the matrix formed by the three vectors.

2. How Does the Calculator Work?

The calculator uses the determinant formula:

\[ a \cdot (b \times c) = \det(\begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{bmatrix}) \]

Which expands to: \[ a_1(b_2c_3 - b_3c_2) - a_2(b_1c_3 - b_3c_1) + a_3(b_1c_2 - b_2c_1) \]

3. Importance of Triple Product Calculation

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

4. Using the Calculator

Tips: Enter all three components for each vector (x, y, z). The calculator will compute the scalar triple product using the determinant method.

5. Frequently Asked Questions (FAQ)

Q1: What does a zero triple product mean?
A: A zero result indicates that the three vectors are coplanar (they all lie in the same plane).

Q2: How is this related to Wolfram Alpha?
A: This calculator implements the same mathematical computation that Wolfram Alpha would perform for a triple product calculation.

Q3: What's the geometric interpretation?
A: The absolute value equals the volume of the parallelepiped formed by the three vectors.

Q4: Can this be negative?
A: Yes, the sign indicates the orientation of the three vectors (right-handed or left-handed system).

Q5: What units should I use?
A: Use consistent units for all components. The result will be in those units cubed (for volume interpretation).