Scalar Triple Product Calculator with Steps

Scalar Triple Product Formula:

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

Vector a (i component):

Vector a (j component):

Vector a (k component):

Vector b (i component):

Vector b (j component):

Vector b (k component):

Vector c (i component):

Vector c (j component):

Vector c (k component):

Scalar Triple Product Result:

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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), where · is the dot product and × is the cross product.

Purpose: It calculates the volume of the parallelepiped formed by the three vectors and determines if the vectors are coplanar (result = 0).

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 calculates the cross product of b and c, then takes the dot product of a with the resulting vector.

3. Importance of Scalar Triple Product

Details: Used in physics, engineering, and computer graphics for volume calculations, determining vector coplanarity, and solving systems of equations.

4. Using the Calculator

Tips: Enter the i, j, k components for each of the three vectors. The calculator will show both the result and the step-by-step calculation.

5. Frequently Asked Questions (FAQ)

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

Q2: Is the order of vectors important?
A: The value remains the same for any cyclic permutation (a·(b×c) = b·(c×a) = c·(a×b)), but changes sign for non-cyclic permutations.

Q3: Can this be used with 2D vectors?
A: No, all three vectors must be 3-dimensional for a meaningful scalar triple product.

Q4: What's the geometric interpretation?
A: The absolute value equals the volume of the parallelepiped formed by the three vectors.

Q5: How is this different from vector triple product?
A: Scalar triple product yields a scalar value, while vector triple product a × (b × c) yields another vector.