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Home Vectors Product of Vectors Inner Product Theorem 2: Algebraic Rules for Inner Products  
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Theorem 2: Algebraic Rules for Inner Products
Here is a list of algebraic rules for inner products. All the rules are easy to prove in either two or three dimensions. THEOREM 2 (Algebraic Rules for Inner Products) (i) A · i = a_{1}, A · j = a_{2}, A · k = a_{3}. (ii) A · 0 = 0 · A = 0. (iii) A · B = B · A (Commutative Law). (iv) A · (B + C) = A · B + A · C (Distributive Law). (v) (tA) · B = t(A · B) (Associative Law). (vi) A · A = A^{2}. PROOF Rule (vi) is proved as follows in three dimensions. A · A = a_{1}a_{1} + a_{2}a_{2} + a_{3}a_{3} = a_{1}^{2} + a_{2}^{2} + a_{3}^{2} = A^{2}. Inner products are useful in the study of perpendicular and parallel vectors.


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