Cross products in R n . Although it is not possible to define a cross product of two vectors in R n as we did for two vectors in R 3 , we can construct a “cross product” of n − 1 vectors in Rn that behaves analogously to the three-dimensional cross product. To be specific, if a1 = (a11, a12, . . . , a1n), a2 = (a21, a22, . . . , a2n), . . . , an−1 = (an−11, an−12, . . . , an−1n) are n − 1 vectors in Rn, we define a1 ×a2 × · · · ×an−1 to be the vector in Rn given by the symbolic determinant
(Here e 1 , . . . , e n are the standard basis vectors for R n .) Exercises 39–42 concern this generalized notion of cross product.
Calculate the following cross product in R4:
(1, 2,−1, 3)×(0, 2,−3, 1)×(−5, 1, 6, 0).
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