Orthogonality can be verified by showing that the vector dot product is zero.
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Sure, let it be the rule that the scalar product of two vectors be denoted with mid-dots and the vector product of two vectors be denoted with a cross.
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The cross product of any vector with itself is zero, so the second term vanishes.
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We certainly couldn't use vector cross product in 2D to handle rotations.
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Orthogonality can be verified by showing the vector dot product is zero.
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A vector space equipped with a scalar product is called an inner product space.
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It uses vectors in combination with the vector dot product.
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However, the seven dimensional cross product does not obey either the vector triple product relationship or the Jacobi identity.
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For torque, the multiplication of force and distance is a vector cross product, while for work it is a dot product.