Let a- b and c be a system of three non-coplanar vectors- Then the system a- b- and c- which satisfies a- a- - b - b- c- c- - 1 and a- b- -a- c-b- a-b- c- - c- a- - c- b- -0 is called the reciprocal system to the vectors a- b and c- a - a- - b - b- - c - c- is a

The correct option is A zero vectorLet #160; a⃗=î b⃗=ĵ c⃗=k̂ By vector reciprocal #160; a⃗'⃗=b⃗×c⃗[a⃗b⃗c⃗] a⃗×a⃗'⃗=a⃗×(b⃗×c⃗)[a⃗b⃗c⃗ ...

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Byju's Answer

Standard XII

Mathematics

First Fundamental Theorem of Calculus

Let a, b an...

Question

Let $a, b$ and $c$ be a system of three non-coplanar vectors. Then the system $a^{'}, b^{'}$ and $c^{'}$ which satisfies $a \cdot a^{'} = b \cdot b^{'} = c \cdot c^{'} = 1$ and $a \cdot b^{'} . = a \cdot c^{'} = b \cdot a^{'} . = b \cdot c^{'} = c \cdot a^{'} = c \cdot b^{'} = 0$ is called the reciprocal system to the vectors $a, b$ and $c$ .
$a \times a^{'} + b \times b^{'} + c \times c^{'}$ is a

zero vector

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a nonzero vector

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\frac{1}{[a b c]^{3}} ((a \times a^{'}) \times b)

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is a scalar multiple of

a^{'} + b^{'} + c^{'}

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Solution

The correct option is A zero vector
Let
$\to a =^i$
$\to b =^j$
$\to c =^k$
By vector reciprocal
$\to a^{'} = \frac{\to b \times \to c}{[\to a \to b \to c]}$
$\to a \times \to a^{'} = \frac{\to a \times (\to b \times \to c)}{[\to a \to b \to c]}$
$\to a \times \to a^{'} = \frac{^i \times (^j \times^k)}{[\to a \to b \to c]}$
$\to a \times \to a^{'} = \frac{0}{[\to a \to b \to c]}$
$\to a \times \to a^{'} = 0$

$\to b^{'} = \frac{\to c \times \to a}{[\to a \to b \to c]}$
$\to b \times \to b^{'} = \frac{\to b \times (\to c \times \to a)}{[\to a \to b \to c]}$
$\to b \times \to b^{'} = \frac{^j \times (^k \times^i)}{[\to a \to b \to c]}$
$\to b \times \to b^{'} = \frac{0}{[\to a \to b \to c]}$
$\to b \times \to b^{'} = 0$

$\to c^{'} = \frac{\to a \times \to b}{[\to a \to b \to c]}$
$\to c \times \to c^{'} = \frac{\to c \times (\to a \times \to b)}{[\to a \to b \to c]}$
$\to c \times \to c^{'} = \frac{^k \times (^i \times^j)}{[\to a \to b \to c]}$
$\to c \times \to c^{'} = \frac{0}{[\to a \to b \to c]}$
$\to c \times \to c^{'} = 0$
$\to a \times \to a^{'} + \to b \times \to b^{'} + \to c \times \to c^{'} = 0$

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Similar questions

Q. Let

a, b

and

c

be a system of three non-coplanar vectors. Then the system

a^{'}, b^{'}

and

c^{'}

which satisfies

a \cdot a^{'} = b \cdot b^{'} = c \cdot c^{'} = 1

and

a \cdot b^{'} . = a \cdot c^{'} = b \cdot a^{'} . = b \cdot c^{'} = c \cdot a^{'} = c \cdot b^{'} = 0

is called the reciprocal system to the vectors

a, b

and

c

a^{'}, b^{'}, c^{'}

is a reciprocal system of

a, b, c

then

Q. Let

a, b

and

c

be a system of three non-coplanar vectors Then the system

a^{'}, b^{'}

and

c^{'}

which satisfies

a \cdot a^{'} = b \cdot b^{'} = c \cdot c^{'} = 1

and

a \cdot b^{'} . = a \cdot c^{'} = b \cdot a^{'} . = b \cdot c^{'} = c \cdot a^{'} = c \cdot b^{'} = 0

is called the reciprocal system to the vectors

a, b

and

c

Find the correct one:

Q. If

A, B, C

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\frac{A \cdot (B \times C)}{(C \times A) \cdot B} + \frac{B \cdot (A \times C)}{C \cdot (A \times B)}

Q. Let

\to a, \to b

and

\to c

be three unit vectors such that

\to a + \to b + \to c = \to 0

. If

λ = \to a \cdot \to b + \to b \cdot \to c + \to c \cdot \to a

and

\to d = \to a \times \to b + \to b \times \to c + \to c \times \to a

, then the ordered pair

(λ, \to d)

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Q. If

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

where

\to a, \to b

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Let a,b and c be a system of three non-coplanar vectors. Then the system a′,b′ and c′ which satisfies a⋅a′=b⋅b′=c⋅c′=1 and a⋅b′.=a⋅c′=b⋅a′.=b⋅c′=c⋅a′=c⋅b′=0 is called the reciprocal system to the vectors a,b and c.a×a′+b×b′+c×c′ is a