VeeEight said:This can get messy, but have you considered the coordinate notation, that is for vectors A = <a1, a2, a3>, B = <b1, b2, b3>, A x B = <a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1>
(http://en.wikipedia.org/wiki/Cross_product#Matrix_notation)
The theorem states that if we have four vectors (A, B, C, and D), and we add them together in a specific order (A + B + C + D), the result will be the same as if we add them in a different order (D + C + B + A).
This theorem is commonly used in physics and engineering to calculate the net force on an object. It is also used in computer graphics to manipulate and transform images.
The first step is to define and understand what a vector is. Then, we can use the properties of vectors (such as commutative and associative properties) to manipulate the equation until we reach the desired result.
This theorem only applies to vector addition. It does not work for other mathematical operations, such as multiplication or division. Additionally, it only applies to a specific set of vectors and may not hold true for all cases.
Sure! Let's say we have four vectors: A = (1, 2), B = (-3, 4), C = (5, -6), and D = (7, 8). If we add them in the order A + B + C + D, we get (1, 2) + (-3, 4) + (5, -6) + (7, 8) = (10, 8). If we add them in a different order, say D + C + B + A, we get (7, 8) + (5, -6) + (-3, 4) + (1, 2) = (10, 8). As you can see, the result is the same, proving the theorem.