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.
Use the generalized notion of cross products to find an equation of the (four-dimensional) hyperplane in R5 through the five points P0(1, 0, 3, 0, 4), P1(2,−1, 0, 0, 5), P2(7, 0, 0, 2, 0), P3(2, 0, 3, 0, 4), and P4(1,−1, 3, 0, 4).
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