Vector Cross Product Rules
The cross product of two vectors yields a third vector in three-dimensional space. The resulting vector is perpendicular to both the cross-multiplied vectors. This mathematical operation is common in engineering equations and vector calculus.
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Calculation
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The cross product is calculated by taking the determinant of the matrix made up of the unit vector symbols in the first row, the "a" vector coefficients in the second row and the "b" coefficients in the third row.
Right-Handed
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By convention, the cross product creates a right-handed system in three-dimensional space. The negative of a cross product creates a left-handed system in three-dimensional space.
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Basis Vectors
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Since the "i," "j" and "k" basis vectors are all mutually perpendicular, each one is the cross product of the other two.
i X j = k
j X k = i
k X i = j
Anticommutative
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In multiplication and addition, the commutative property allows the reversal of the terms. For instance, a x b = b x a. A vector cross product is "anticommutative," which means a X b = -(b X a).
Scalar multiplication
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Any scalar multiplied into a cross product has the following property:
k(a X b) = (ka) X b = a X (kb), where a and be are vectors and k is a scalar. This is similar to the distributive property in multiplication, except k is distributed to either a or b and not both.
Distributive
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If a vector sum is part of a cross product, the cross product can be distributed, but the order of the cross product must be preserved:
a X (b + c) = (a X b) + (a X c)
(a + b) X c = (a X c) + (b X c)
where a, b and c are all vectors.
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References
- Photo Credit Vector cross product
