Scalar Triple Product Calculator Calculus

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:

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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 results in a scalar value.

Purpose: It calculates the volume of the parallelepiped formed by the three vectors and is useful in vector calculus and physics applications.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

a, b, c — Three-dimensional vectors
× — Cross product operation
· — Dot product operation

Explanation: First calculates the cross product of b and c, then takes the dot product of that result with vector a.

3. Importance of Scalar Triple Product

Details: The scalar triple product gives the signed volume of the parallelepiped formed by the three vectors. It's zero when the vectors are coplanar.

4. Using the Calculator

Tips: Enter the i, j, and k 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 indicate?
A: The sign indicates the orientation of the three vectors (right-handed or left-handed system).

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

Q3: Can I use this for 2D vectors?
A: No, scalar triple product requires three 3D vectors. For 2D, you would need to add a zero third component.

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 used in physics?
A: It's used in calculating torques, angular momentum, and in fluid mechanics calculations.