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Sum of Coefficients of All Terms

If a⃗,b⃗,c⃗...

Question

If $\to a, \to b, \to c$ and $\to d$ are unit vectors such that $(\to a \times \to b) \cdot (\to c \times d) = 1$ and $\to a \cdot \to c = \frac{1}{2}$ , then

A

\to a, \to b, \to c

are non-coplanar

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B

\to b, \to c, \to d

are non-coplanar

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C

\to b, \to d

are non-parallel

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D

\to a, \to d

are parallel and

\to b, \to c

are parallel

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Solution

The correct option is C $\to b, \to d$ are non-parallel
$(\to a \times \to b) \cdot (\to c \times d) = 1$

$\Rightarrow (\to a . \to c) (\to b . \to d) - (\to a . \to d) (\to b . \to c) = 1$

$\Rightarrow \frac{1}{2} (\to b . \to d) - (\to a . \to d) (\to b . \to c) = 1$

$\Rightarrow \to b . \to d = 2 [1 - (\to a . \to d) (\to b . \to c)] \geq 0$ , since $a, b, c, d$ are unit vector
Hence $\to b, \to d$ are non-parallel.

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Similar questions

\to b, \to c, \to d

are non-coplanar vectors, then the vector

(\to a \times \to b) \times (\to c \times \to d) + (\to a \times \to c) \times (\to d \times \to b) + (\to a \times \to d) \times (\to b \times \to c)

\to b, \to c, \to d

are non-coplanar vectors then the vector

(\to a \times \to b) \times (\to c \times \to d) + (\to a \times \to c) \times (\to d \times \to b) + (\to a \times \to d) \times (\to b \times \to c)

\to a, \to b, \to c, \to d

are non-coplanar vectors then the vector

(\to a \times \to b) \times (\to c \times \to d) + (\to a \times \to c) \times (\to d \times \to b) + (\to a \times \to d) \times (\to b \times \to c)

is parallel to:

If $\to a, \to b, \to c$ , and $\to d,$ are the unit vectors such that $(\to a \times \to b) . (\to c \times \to d) = 1$ and $\to a . \to c = \frac{1}{2},$ then

\to a, \to b, \to c

\to d

are unit vectors such that

(\to a \times \to b) . (\to c \times \to d) = λ

\to a . \to c = \frac{\sqrt{3}}{2}

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