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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 a × b is defined as a vector c that is perpendicular (orthogonal) to both a and b, with a direction given by the right-hand rule [1] and a magnitude equal to the area of the parallelogram that the vectors span. [2] The cross product is defined by the formula [8] [9]
On the other hand, a ... Cross product rule ... The figure to the right is a mnemonic for some of these identities. The abbreviations used are:
The vector product and the scalar product of two vectors define the angle between them, say θ: [1] [3] = ‖ ‖ ‖ ‖ ‖ ‖ (<) To satisfy the right-hand rule, for positive θ, vector B is counter-clockwise from A, and for negative θ it is clockwise.
For an oriented surface, the normal is usually determined by the right-hand rule or its analog in higher dimensions. If the normal is constructed as the cross product of tangent vectors (as described in the text above), it is a pseudovector.
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The rule of three [1] was a historical shorthand version for a particular form of cross-multiplication that could be taught to students by rote. It was considered the height of Colonial maths education [ 2 ] and still figures in the French national curriculum for secondary education, [ 3 ] and in the primary education curriculum of Spain.