Vector Product of VectorsThe vector product and the scalar product are the two ways of multiplying vectors which see the most application in physics and astronomy. The magnitude of the vector product of two vectors can be constructed by taking the product of the magnitudes of the vectors times the sine of the angle (<180 degrees) between them. The magnitude of the vector product can be expressed in the form: and the direction is given by the righthand rule. If the vectors are expressed in terms of unit vectors i, j, and k in the x, y, and z directions, then the vector product can be expressed in the rather cumbersome form: which may be stated somewhat more compactly in the form of a determinant. 
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Vector Product, Determinant FormThe vector product is compactly stated in the form of a determinant: Once the scheme for determinant evaluation is familiar, this is a convenient way to reconstruct the expanded form: 
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Vector Product CalculationYou may enter values in any of the boxes below. Then click on the symbol for either the vector product or the angle. 
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Vector Product ApplicationsGeometrically, the vector product is useful as a method for constructing a vector perpendicular to a plane if you have two vectors in the plane. Physically, it appears in the calculation of torque and in the calculation of the magnetic force on a moving charge. 
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Right Hand Rule, Vector ProductThe direction of the vector product can be visualized with the righthand rule. If you curl the fingers of your right hand so that they follow a rotation from vector A to vector B, then the thumb will point in the direction of the vector product. The vector product of A and B is always perpendicular to both A and B. Another way of stating that is to say that the vector product is perpendicular to the plane formed by vectors A and B. This righthand rule direction is produced mathematically by the vector product expression. 
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