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Addition of Vectors

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Question

Let $\to a \times (\to b \times \to c) = \frac{\to b}{3} + \frac{\to c}{2}$ and $\to b \times (\to c \times \to a) = - \frac{\to c}{2}$
If $\to a, \to b$ and $\to c$ are non-collinear pair wise unit vectors, then volume of a parallelopiped, whose coterminous edges are $\to a, \to b$ and $\to c$ , is

A

\frac{11}{36}

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B

\frac{1}{2}

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C

\frac{23}{36}

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D

None of these

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Solution

The correct option is D None of these
$\to a \times (\to b \times \to c) = \frac{\to b}{3} + \frac{\to c}{2}$
$\Rightarrow (\to a . \to c) \to b - (\to a . \to b) \to c = \frac{\to b}{3} + \frac{\to c}{2}$
$\to a . \to c = \frac{1}{3}$ & $\to a . \to b = - \frac{1}{2}$
$\to b \times (\to c \times \to a) = - \frac{\to c}{2}$
$\Rightarrow (\to b . \to a) \to c - (\to b . \to c) \to a = - \frac{\to c}{2} \Rightarrow \to b . \to c = 0$
volume of parallelopiped $= | [\to a \to b \to c] |$
$[\to a \to b \to c]^{2} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} \to a . \to a & \to a . \to b & \to a . \to c \to b . \to a & \to b . \to b & \to b . \to c \to c . \to a & \to c . \to b & \to c . \to c \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & - \frac{1}{2} & \frac{1}{3} - \frac{1}{2} & 1 & 0 \frac{1}{3} & 0 & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ = \frac{23}{36}$
$∴ volume = \frac{\sqrt{23}}{6}$

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Similar questions

\to a, \to b, \to c

are unit vectors and

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are non-collinear vectors satisfying

(\to a, \to b) = α, (\to a, \to c) = β

\to a \times (\to b \times \to c) = \frac{\to b + \to c}{2}

c o s (α + β) =

Q. If the volume of a Tetrahedron formed by coterminous edges

\to a, \to b, \to c

2

, then the volume of the parallelopiped formed by the coterminous edges

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

\to a

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\to c

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

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