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(D) We are given four vectors $a, b, c$, and $d$ such that $a \cdot b = 0$, $|a 	imes c| = |a||c|$, and $|a 	imes d| = |a||d|$.From the conditions $

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(D) We are given four vectors $a, b, c$, and $d$ such that $a \cdot b = 0$, $|a 	imes c| = |a||c|$, and $|a 	imes d| = |a||d|$.From the conditions $|a 	imes c| = |a||c|$ and $|a 	imes d| = |a||d|$, we know that $\sin 	heta = 1$ for the angles between $a, c$ and $a, d$ respectively.This implies that $a \perp c$ and $a \perp d$.Since both $c$ and $d$ are perpendicular to $a$, the vector $(c 	imes d)$ must be parallel to $a$.Therefore, we can write $c 	imes d = \lambda a$ for some scalar constant $\lambda$.Now, we evaluate the scalar triple product $[b c d] = b \cdot (c 	imes d)$.Substituting $c 	imes d = \lambda a$, we get $[b c d] = b \cdot (\lambda a) = \lambda (b \cdot a)$.Since $a \cdot b = 0$, it follows that $b \cdot a = 0$.Thus, $[b c d] = \lambda 	imes 0 = 0$.

If $a, b, c, d$ are $4$ vectors such that $a \cdot b = 0$, $|a \times c| = |a||c|$, and $|a \times d| = |a||d|$, then $[b c d] = $

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