Triple Vector Cross Product Calculator 2 Vectors

Triple Cross Product Identity:

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

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):

Triple Cross Product Result:

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1. What is the Triple Vector Cross Product?

The triple vector cross product identity relates three vectors in 3D space. The identity states that (a × b) × c = b(a·c) - c(a·b), where × denotes the cross product and · denotes the dot product.

2. How Does the Calculator Work?

The calculator uses the triple cross product identity:

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

Where:

\( a, b, c \) — Input vectors in 3D space
\( \times \) — Cross product operation
\( \cdot \) — Dot product operation

Explanation: The identity allows calculation of the cross product of a cross product (a × b) with a third vector c using only dot products.

3. Importance of Triple Cross Product

Details: This identity is fundamental in vector calculus and physics, particularly in mechanics and electromagnetism. It simplifies complex vector operations and helps solve problems involving multiple vector products.

4. Using the Calculator

Tips: Enter the x, y, z components of all three vectors. The calculator will compute the triple cross product using the vector identity.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between (a × b) × c and a × (b × c)?
A: They are different operations with different results. The vector triple product is not associative.

Q2: What are some applications of this identity?
A: Used in physics for torque calculations, angular momentum, and electromagnetic field computations.

Q3: Does this work in 2D space?
A: No, the cross product is only defined in 3D space (and 7D with a different definition).

Q4: What if two vectors are parallel?
A: If a and b are parallel, a × b = 0, so (a × b) × c = 0.

Q5: Can I use this for non-vector quantities?
A: No, this identity specifically applies to vectors in three-dimensional space.