Vector Triple Product Calculator Calculus

Vector Triple Product Formula:

\[ \vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c} \]

Vector a (x):

Vector a (y):

Vector a (z):

Vector b (x):

Vector b (y):

Vector b (z):

Vector c (x):

Vector c (y):

Vector c (z):

Vector Triple Product Result:

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1. What Is The Vector Triple Product?

The vector triple product is a mathematical operation that takes three vectors and produces a new vector. The formula \( \vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c} \) shows that the result lies in the plane spanned by vectors b and c.

2. How Does The Calculator Work?

The calculator uses the vector triple product formula:

\[ \vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c} \]

Where:

\( \vec{a} \cdot \vec{c} \) — Dot product of vectors a and c
\( \vec{a} \cdot \vec{b} \) — Dot product of vectors a and b
\( \vec{b} \) — Vector b components
\( \vec{c} \) — Vector c components

Explanation: The calculator first computes the dot products, then scales vectors b and c by these values, and finally subtracts them to get the result.

3. Applications Of Vector Triple Product

Details: The vector triple product is used in physics, engineering, and computer graphics for calculating torques, angular momentum, and solving problems involving rotational motion and vector projections.

4. Using The Calculator

Tips: Enter the x, y, and z components for all three vectors. The calculator will compute the vector triple product using the formula and display the resulting vector.

5. Frequently Asked Questions (FAQ)

Q1: What is the geometric interpretation of the vector triple product?
A: The result is perpendicular to vector a and lies in the plane containing vectors b and c. Its magnitude represents the volume of the parallelepiped formed by the three vectors.

Q2: Does the vector triple product follow the associative property?
A: No, the vector triple product is not associative. \( \vec{a} \times (\vec{b} \times \vec{c}) \neq (\vec{a} \times \vec{b}) \times \vec{c} \) in general.

Q3: What are some practical applications of this formula?
A: It's used in physics for calculating magnetic forces, in computer graphics for lighting calculations, and in engineering for stress analysis and fluid dynamics.

Q4: Can this calculator handle 2D vectors?
A: Yes, for 2D vectors, simply enter 0 for the z-component of all vectors.

Q5: What is the relationship between the vector triple product and the scalar triple product?
A: The scalar triple product \( \vec{a} \cdot (\vec{b} \times \vec{c}) \) gives a scalar value representing volume, while the vector triple product gives a vector result.