why cross product not follow the law of associative law
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the cross product or vector product(occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space {\displaystyle \left(\mathbb {R} ^{3}\right)} and is denoted by the symbol {\displaystyle \times }. Given two linearly independent vectors {\displaystyle \mathbf {a} } and {\displaystyle \mathbf {b} }, the cross product, {\displaystyle \mathbf {a} \times \mathbf {b} }, is a vector that is perpendicular to both {\displaystyle \mathbf {a} } and {\displaystyle \mathbf {b} } and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product).
If two vectors have the same direction (or have the exact opposite direction from one another, i.e. are not linearly independent) or if either one has zero length, then their cross product is zero. More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths. The cross product is anticommutative (i.e., {\displaystyle \mathbf {a} \times \mathbf {b} =-\mathbf {b} \times \mathbf {a} }) and is distributive over addition (i.e., {\displaystyle \mathbf {a} \times (\mathbf {b} +\mathbf {c} )=\mathbf {a} \times \mathbf {b} +\mathbf {a} \times \mathbf {c} }). The space {\displaystyle \mathbb {R} ^{3}} together with the cross product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket.
Like the dot product, it depends on the metric of Euclidean space, but unlike the dot product, it also depends on a choice of orientation or "handedness". The product can be generalized in various ways; it can be made independent of orientation by changing the result to pseudovector, or in arbitrary dimensions the exterior product of vectors can be used with a bivector or two-form result. Also, using the orientation and metric structure just as for the traditional 3-dimensional cross product, one can in {\displaystyle n} dimensions take the product of {\displaystyle n-1}vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions.[1] If one adds the further requirement that the product be uniquely defined, then only the 3-dimensional cross product qualifies. (See § Generalizations, below, for other dimensions.)
If two vectors have the same direction (or have the exact opposite direction from one another, i.e. are not linearly independent) or if either one has zero length, then their cross product is zero. More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths. The cross product is anticommutative (i.e., {\displaystyle \mathbf {a} \times \mathbf {b} =-\mathbf {b} \times \mathbf {a} }) and is distributive over addition (i.e., {\displaystyle \mathbf {a} \times (\mathbf {b} +\mathbf {c} )=\mathbf {a} \times \mathbf {b} +\mathbf {a} \times \mathbf {c} }). The space {\displaystyle \mathbb {R} ^{3}} together with the cross product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket.
Like the dot product, it depends on the metric of Euclidean space, but unlike the dot product, it also depends on a choice of orientation or "handedness". The product can be generalized in various ways; it can be made independent of orientation by changing the result to pseudovector, or in arbitrary dimensions the exterior product of vectors can be used with a bivector or two-form result. Also, using the orientation and metric structure just as for the traditional 3-dimensional cross product, one can in {\displaystyle n} dimensions take the product of {\displaystyle n-1}vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions.[1] If one adds the further requirement that the product be uniquely defined, then only the 3-dimensional cross product qualifies. (See § Generalizations, below, for other dimensions.)
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