Vectors A and B satisfying the vector equation A-B-a- A-B-b and A-a-1- where a and b are given vectors- are

We have A+B=a or A·a+B·a=a·a or 1+B· nbsp;a=a^2 nbsp; (given A·a=1) or B·a=a^2 1 nbsp; nbsp;(i) Also, A× nbsp;B=b or a×(A×B)=a×b or (a·B)A (a·A)B=a×b ...

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Standard XI

Mathematics

Applications of Cross Product

Vectors A and...

Question

Vectors $\to A$ and $\to B$ satisfying the vector equation $\to A + \to B = \to a, \to A \times \to B = \to b$ and $\to A \cdot \to a = 1$ , where $\to a$ and $\to b$ are given vectors, are

\to A = \frac{(\to a \times \to b) - \to a}{| \to a |^{2}}

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\to B = \frac{(\to b \times \to a) + \to a (| \to a |^{2} - 1)}{| \to a |^{2}}

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\to A = \frac{(\to a \times \to b) + \to a}{| \to a |^{2}}

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\to B = \frac{(\to b \times \to a) - \to a (| \to a |^{2} - 1)}{| \to a |^{2}}

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Solution

The correct option is C $\to A = \frac{(\to a \times \to b) + \to a}{| \to a |^{2}}$
We have $\to A + \to B = \to a$
or $\to A \cdot \to a + \to B \cdot \to a = \to a \cdot \to a$
or $1 + \to B \cdot \to a = a^{2}$ (given $\to A \cdot \to a = 1$ )
or $\to B \cdot \to a = a^{2} - 1$ $(i)$
Also, $\to A \times \to B = \to b$
or $\to a \times (\to A \times \to B) = \to a \times \to b$
or $(\to a \cdot \to B) \to A - (\to a \cdot \to A) \to B = \to a \times \to b$
or $(a^{2} - 1) \to A - \to B = \to a \times \to b$ $(i i)$
(using $(i)$ and $\to a \cdot \to A = 1$ )
and $\to A + \to B = a$ $(i i i)$
From $(i i)$ and $(i i i)$ , we have
$\to A = \frac{(\to a \times \to b) + \to a}{a^{2}}$
$\to B = \to a - ⎧ ⎪ ⎨ ⎪ ⎩ \frac{(\to a \times \to b) + \to a}{a^{2}} ⎫ ⎪ ⎬ ⎪ ⎭$
$\to B = \frac{(\to b \times \to a) + \to a (a^{2} - 1)}{a^{2}}$

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Vectors →A and →B satisfying the vector equation →A+→B=→a, →A×→B=→b and →A⋅→a=1, where →a and →b are given vectors, are

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