Here ê, are orthonormal unit base vectors in R³. 2.12 Let the vectors (i, j, k) constitute an orthonormal basis. In terms of this basis, define a cogredient basis by e₁=-Î-Ĵ, e₂ e2 = 1+2ĵ-2k, e3 e3 = 2î+j+ k. = Determine (a) the dual or reciprocal (contragredient) basis (e¹,e², e³) in terms of the orthonormal basis (î, j, k), (b) the magnitudes (or norms) |e₁|, |e2|, |e3|, |e¹|, |e²|, and |e³|, and (c) the cogredient components A1, A2, and A3 of a vector A if its contragredient components are given by A¹ = 1, A² = 2, A³ = 3.

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