Prove i × j = k and k × j = −i by each of the following methods:
(a) Using the definition of cross product as a determinant
(b) Using the geometric description of the cross product in Theorem 1

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THEOREM 1 Geometric Description of the Cross Product Given two nonzero non- parallel vectors v and w with angle between them, the cross product v x w is the unique vector with the following three properties: (i) vx wis orthogonal to v and w. (ii) v x w has length ||v||||w|| sin 8. (iii) {v, w, vx w} forms a right-handed system.