Cross products in R n . Although it is not possible to define a cross prod...

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, . . . , a_1n), a₂ = (a₂₁, a₂₂, . . . , a_2n), . . . , a_n−1 = (a_n−11, a_n−12, . . . , a_n−1n) are n − 1 vectors in Rⁿ, we define a₁ ×a₂ × · · · ×a_n−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).