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Acceleration

If a⃗,b⃗, c...

Question

If $\to a, \to b, \to c$ are non-caplanar non-zero vectors and $\to r$ is any vector in space then $[\to b \to c \to r] \to a + [\to c \to a \to r] \to b + [\to a \to b \to r] \to c$ is equal to

A

3 [\to a \to b \to c] \to r

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B

[\to a \to b \to c] \to r

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C

[\to b \to c \to a] \to r

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D

[\to c \to a \to b] \to r

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Solution

The correct options are
B $[\to a \to b \to c] \to r$
C $[\to b \to c \to a] \to r$
D $[\to c \to a \to b] \to r$

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Similar questions

\to a, \to b, \to c

are three non-coplanar non-zero vectors and

\to r

is any vector, then

(\to a \times \to b) \times (\to r \times \to c) + (\to b \times \to c) \times (\to r \times \to a) + (\to c \times \to a) \times (\to r \times \to b) =

\to a, \to b, \to c

are non coplanar vectors and

\to p, \to q, \to r

\to p = \frac{\to b \times \to c}{[\begin{matrix} \to b & \to c & \to a \end{matrix}]}

\to q = \frac{\to c \times \to a}{[\begin{matrix} \to c & \to a & \to b \end{matrix}]}

\to r = \frac{\to a \times \to b}{[\begin{matrix} \to a & \to b & \to c \end{matrix}]}

(\to a + \to b) . \to p + (\to b + \to c) . \to q + (\to c + \to a) . \to r

¯ ¯ ¯ a, ¯ ¯ b, ¯ ¯ c

are three non-coplanar vectors,

¯ ¯ ¯ p, ¯ ¯ ¯ q, ¯ ¯ ¯ r

are non-zero vectors such that

\to p = \frac{\to b \times \to c}{[\begin{matrix} \to a & \to b & \to c \end{matrix}]}, \to q = \frac{\to c \times \to a}{[\begin{matrix} \to a & \to b & \to c \end{matrix}]}, \to r = \frac{\to a \times \to b}{[\begin{matrix} \to a & \to b & \to c \end{matrix}]}

, then the value of the expression

(\to a + \to b + \to c) . (\to p + \to q + \to r)

\to a, \to b, \to c

be three non-coplanar vectors and

\to r

be any arbitrary vector, then

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

\to a, \to b, \to c

are three noncoplanar nonzero vectors and

\to r

is any vector in space then

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

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