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Multiplication with Vectors

If A⃗, B⃗ ...

Question

If $\to A$ , $\to B$ & $\to C$ are vectors such that $| \to B | =$ | $\to C |$ , Prove that : $[(\to A + \to B) \times (\to A + \to C)] \times (\to B \times \to C) . (\to B + \to C) = 0$

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Solution

$\begin{matrix} {\to A \times \to C + \to B \times \to A + \to B \times \to C} \times {(\to B \times \to C)} . (\to B + \to C) \Rightarrow (\to A \times \to C + \to B \times \to A + \to B \times \to C) . ((\to B \times \to C) \times (\to B + \to C)) \Rightarrow (\to A \times \to C + \to B \times \to A + \to B \times \to C) . ((B^{2}) \to C + (\to B . \to C) \to C - (\to B . \to C) \to B - (C^{2}) \to B) - [A C B] (\to B \cdot \to C - C^{2}) + [B A C] (\to B + \to B . \to C) - [A C B] (\to B \cdot \to C) - [A C B] C^{2} + [B A C] B^{2} + [B A C] \to B . \to C = 0 (∣ ∣ \to B ∣ ∣ = ∣ ∣ \to C ∣ ∣) \end{matrix}$

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Similar questions

\to A, \to B

\to C

are vectors such that

| \to B | = | \to C |

[(\to A + \to B) \times (\to A + \to C)] \times (\to B \times \to C) . (\to B + \to C) = 0

\to a

\to b

\to c

are three vectors and

\to a + \to b + \to c = 0

\to a \times \to b = \to b \times \to c = \to c \times \to a

\to a, \to b, \to c

are three vectors such that

\to a \times \to b = \to c, \to b \times \to c = \to a, \to c \times \to a = \to b

then prove that

| \to a | = | \to b | = | \to c |

\to a, \to b, \to c

are three vectors and

| \to b | = | \to c |

(\to a + \to b) \times (\to a + \to c)} \times (\to b \times \to c) . (\to b + \to c) =

\to a, \to b, \to c

are three vectors such that

\to a, \to b, \to c

, then prove that

\to a \times \to b = \to b \times \to c = \to c \times \to a

, and hence show that

[\to a \to b \to c] = 0.

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