Question

One of the following laws for dot and cross multiplication of three vectors A , B and C is invalid:

(A.B)C=A(B.C)

A×(B×C)=(A.C)B−(A.B)C

(A×B)×C=(A.C)B−(B.C)A

A×(B×C)≠(A×B)×C

Answer 1

Answer on Question #56040 – Math – Vector Calculus

One of the following laws for dot and cross multiplication of three vectors A, B and C is invalid:

$$(A.B)C = A(B.C)$$ $$A \times (B \times C) = (A.C)B - (A.B)C$$ $$(A \times B) \times C = (A.C)B - (B.C)A$$ $$A \times (B \times C) \neq (A \times B) \times C$$

Solution

$$(A.B)C = A(B.C)$$ $$(\bar{A} \cdot \bar{B}) \bar{C} = \bar{A} (\bar{B} \cdot \bar{C}),$$

The dot products are scalars, so it means

$$\bar{C} = k \bar{A}, \text{ where } k = \frac{(\bar{B} \cdot \bar{C})}{(\bar{A} \cdot \bar{B})}.$$

So $\bar{C}$ is some scalar multiple of $\bar{A}$. Thus, this rule is not true for any three vectors A, B and C.

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