Cross product is a mathematics term when there is a binary operation on two vectors in three-dimensional space.
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The cross product can be said to be a measure of the 'perpendicularity' of the vectors in the product. Please see the link.
Normally you use sine theta with the cross product and cos theta with the vector product, so that the cross product of parallel vectors is zero while the dot product of vectors at right angles is zero.
A dot product is a scalar product so it is a single number with only one component. A cross product or vector product is a vector which has three components like the original vectors.
No. A cross product is just a way of simplifying a proportion. If the cross product aren't equal, it follows logically that the proportion isn't equal.