If a is parallel to b×c then (a×b)·(a×c)=

a || b×c ⇒ a⊥b, a⊥c ⇒ a·b=0, a·c=0(a×b)·(a×c)=a·aa·cb·ab·c =|a|200b·c=|a|2 (b·c)

If a is parallel to b×c then (a×b)·(a×c)=

a || b×c ⇒ a⊥b, a⊥c ⇒ a·b=0, a·c=0(a×b)·(a×c)=a·aa·cb·ab·c =|a|200b·c=|a|2 (b·c)

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If $a$ is parallel to $b \times c$ then $(a \times b) \cdot (a \times c) =$

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$a^{2} (b \cdot c)$

$b^{2} (a \cdot c)$

$c^{2} (a \cdot b)$

answer is A.

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Detailed Solution

$\begin{array}{rcl} a | | b \times c & \Rightarrow & a ⊥ b, a ⊥ c \\ \Rightarrow & a \cdot b = 0, a \cdot c = 0 \\ (a \times b) \cdot (a \times c) & = & |\begin{matrix} a \cdot a & a \cdot c \\ b \cdot a & b \cdot c \end{matrix}| \\ = & |\begin{matrix} | a |^{2} & 0 \\ 0 & b \cdot c \end{matrix}| = | a |^{2} (b \cdot c) \end{array}$