Question

Simplify
(A+B).(B+C)×(C+A)

a). B×C

b). 2.A×B

c). 2.A×C

d). A×B×C

Answer 1

Answer on Question #56115 – Math - Vector Calculus

Question

Simplify

$$(A + B). (B + C) \times (C + A)$$

a). $B \times C$

b). $2. A \times B$

c). $2. A \times C$

d). $A \times B \times C$

Solution

By the properties of cross product, calculate

$$\begin{array}{l} (\vec{A} + \vec{B}) \cdot \left((\vec{B} + \vec{C}) \times (\vec{C} + \vec{A})\right) = (\vec{A} + \vec{B}) \cdot (\vec{B} \times \vec{C} + \vec{B} \times \vec{A} + \vec{C} \times \vec{C} + \vec{C} \times \vec{A}) \\ = (\vec{A} + \vec{B}) \cdot (\vec{B} \times \vec{C} + \vec{B} \times \vec{A} + 0 + \vec{C} \times \vec{A}) \\ = \vec{A} \cdot \vec{B} \times \vec{C} + \vec{A} \cdot \vec{B} \times \vec{A} + \vec{A} \cdot \vec{C} \times \vec{A} + \vec{B} \cdot \vec{B} \times \vec{C} + \vec{B} \cdot \vec{B} \times \vec{A} + \vec{B} \cdot \vec{C} \times \vec{A} \\ = \vec{A} \cdot \vec{B} \times \vec{C} + 0 + 0 + 0 + 0 + \vec{B} \cdot \vec{C} \times \vec{A} = 2 \cdot \vec{A} \cdot \vec{B} \times \vec{C}. \end{array}$$

Answer. $2 \cdot \vec{A} \cdot \vec{B} \times \vec{C}$

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