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Right-hand rule for cross product. The cross product of vectors and is a vector perpendicular to the plane spanned by and with the direction given by the right-hand rule: If you put the index of your right hand on and the middle finger on , then the thumb points in the direction of .
The cross product with respect to a right-handed coordinate system. In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here ), and is denoted by the symbol .
2.7 Cross product rule. ... and points in the direction of, the function's most rapid (positive) change. ... for the outer product of two vectors, ...
Dually, one can express the 'meet', or intersection of two straight lines by the cross-product: x ∝ l × m {\displaystyle \mathbf {x} \propto \mathbf {l} \times \mathbf {m} } The relationship to Plücker matrices becomes evident, if one writes the cross product as a matrix-vector product with a skew-symmetric matrix:
The following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.
Here the 1×3 vector x represents an arbitrary point on the line through particular point a with b representing the direction of the line and with the value of the real number determining where the point is on the line, and similarly for arbitrary point y on the line through particular point c in direction d. The cross product of b and d is ...
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There are two lists of mathematical identities related to vectors: Vector algebra relations — regarding operations on individual vectors such as dot product, cross product, etc. Vector calculus identities — regarding operations on vector fields such as divergence, gradient, curl, etc.