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Tension in a String

Identify the ...

Question

Identify the statement (s) which is/are INCORRECT?

A

\to a \times [\to a \times (\to a \times \to b)] = (\to a \times \to b) ({\to a}^{2})

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B

If

\to a, \to b, \to c

are non coplanar vectors and

\to v . \to a = \to v . \to b = \to v . \to c = 0

, then

\to v

must be a null vector

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C

If

\to a

and

\to b

lie in a plane normal to the plane containing the vectors

\to c

and

\to d

, then

(\to a \times \to b) \times (\to c \times \to d) = 0

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D

If

\to a, \to b, \to c

and

^{'},^{'},^{'}

are reciprocal system of vectors, then

\to a .^{'} + \to b .^{'} + \to c .^{'} = 3

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Solution

The correct options are
A $\to a \times [\to a \times (\to a \times \to b)] = (\to a \times \to b) ({\to a}^{2})$
C If $\to a$ and $\to b$ lie in a plane normal to the plane containing the vectors $\to c$ and $\to d$ , then $(\to a \times \to b) \times (\to c \times \to d) = 0$
D If $\to a, \to b, \to c$ and $^{'},^{'},^{'}$ are reciprocal system of vectors, then $\to a .^{'} + \to b .^{'} + \to c .^{'} = 3$
A) $\to a \times [a \times (\to a \times \to b)] = \to a \times [(a . b) \to a - (\to a . \to a) \to b] = 0 - {(\to a)}^{2} (\to a \times \to b)$ False
B) $\to a, \to b, \to c$ are non-coplanar
$\begin{matrix} \to v \to a = 0 \to v \to b = 0 \to v \to c = 0 \end{matrix} ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ \Rightarrow \to v (\to a + \to b + \to c) = 0$ But $\to a + \to b + \to c \neq 0 \Rightarrow \to v = 0$ i.e. null vector which is true
C) $\to a \times \to b$ & $\to c \times \to d$ are perpendicular so $(\to a \times \to b) \times (\to c \times \to d) \neq 0$ False
D) $a^{'} = \frac{\to b \times \to c}{[a b c]}, b^{'} = \frac{\to c \times \to a}{[a b c]}, c^{'} = \frac{\to a \times \to b}{[a b c]}$ is valid only if $\to a, \to b, \to c$ are non coplanar,

The scalar product of any vector of one system with a vector of other system which does not correspond to it is zero
$∴ \to a . {\to b}^{'} + \to b . {\to c}^{'} + \to c . {\to a}^{'} = 0$
Hence false.

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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 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)

\to a

\to b

are two perpendicular unit vectors then

\to a \times (\to a \times \to b)

Q. Given three vectors

\to a, \to b

\to c

, each two of which are non collinear (Equally Separated). Further if

(\to a + \to b)

is collinear with

\to c, (\to b + \to c)

is collinear with

\to a

| \to a | = ∣ ∣ \to b ∣ ∣ = | \to c | = \sqrt{2}

then the value of

\to a . \to b + \to b . \to c + \to c . \to a

:

\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}

Q. If the vectors

\to a

\to b

\to c

\to d

are coplanar, then

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

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