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Perhaps the above are overly-complicated ways to evaluate the triple cross product, but demonstrate different ways to view the expression.
en.wikipedia.org
Because the cross product may also be a (true) vector, it may not change direction with a mirror image transformation.
en.wikipedia.org
This leads to widespread use in physics, wherever the cross product occurs.
en.wikipedia.org
Other examples of pseudovectors include magnetic field, torque, or more generally any cross product of two (true) vectors.
en.wikipedia.org
This may help you to remember the correct cross product formula.
en.wikipedia.org
The cross product and triple product in three dimensions each admit both geometric and algebraic interpretations.
en.wikipedia.org
This condition determines the magnitude of the cross product.
en.wikipedia.org
A similar convention distinguishes between the cross product and the dot product of two vectors.
en.wikipedia.org
The definitions of torque and angular momentum also involve the cross product.
en.wikipedia.org
This can also be used to define the cross product.
en.wikipedia.org

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