What is the derivative of cross product?


Quick Answer

The derivative of a cross product is found through the cross product formula Dt(r(t)×q(t))=r?(x)×q(x)+r(x)×q?(x), where r and q are the cross product values. The derivative is found by plugging in the known cross product values into the formula and solving for x.

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Full Answer

The cross product formula is commonly used to find a vector that is orthogonal to two other vectors, both of which have been computed into the formula. It is also used to find areas of triangles or parallelograms that are formed by the initial two vectors as they join together.

If the vectors have the same direction or one has a length equal to zero, the cross product is zero. The magnitude of the product equals the area of a parallelogram with the vectors for sides. For perpendicular vectors, this is a rectangle, and the magnitude of the product is the product of their lengths. The cross product is not commutative and is distributive.

The Derivation of the cross product formula can also be expressed as such:

For a = (a1/a2/a3) and b = (b1/b2/b3)

a x b = ||a|| x ||b|| sin n (cap)= (a2b3 - a3b2 / a3b1 - a1b3 / a1b2 - a2b1)

The a and b are nonequivalent vectors with different directions, and n (cap) is a unit vector orthogonal to both a and b.

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