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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 .
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 in respect to a right-handed coordinate system. Calculate cross products, take the cross products of the observational unit direction ...
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.
In three dimensions the cross product is invariant under the action of the rotation group, SO(3), so the cross product of x and y after they are rotated is the image of x × y under the rotation. But this invariance is not true in seven dimensions; that is, the cross product is not invariant under the group of rotations in seven dimensions, SO(7).
Here F is the force on the particle, q is the particle's electric charge, E is the external electric field, v, is the particle's velocity, and × denotes the cross product. The direction of force on the charge can be determined by a mnemonic known as the right-hand rule (see the figure).
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The direction of ω is chosen using the right-hand rule. With this convention for depicting rotation, the velocity is given by a vector cross product as =, which is a vector perpendicular to both ω and r(t), tangential to the orbit, and of magnitude ω r.