Cross Product of Vectors

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This Demonstration computes and displays the cross product (black) of two vectors (red) and (blue) in three dimensions. The dot product of the vectors is a scalar (number), while the cross product is a vector.


The cross product can be defined in several equivalent ways.

Geometrically: (1) The length of the vector is given by , where is the angle between and . (The length is equal to the area of the parallelogram spanned by the vectors and .) (2) The direction of , when , is perpendicular to both and , oriented in the sense that , , form a right-handed system.

abcdefAlgebraically: In Cartesian coordinates, the components of the cross product can be read off a determinant, w=|⁠ijkaInlineMathbInlineMathcInlineMathdInlineMatheInlineMathfInlineMath⁠|, where , , are the Cartesian unit vectors and , .


Contributed by: S. M. Blinder and Amy Blinder (March 2011)
Open content licensed under CC BY-NC-SA



Snapshot 1: when and are in the , plane, has only a component

Snapshot 2: this shows the unit vector relationship

Snapshot 3: when and are collinear, their cross product vanishes

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