If 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- then which of the following is-are true

We have A+B=a Taking dot product of a both sides, we get A·a+ B·a=a·a ⇒ 1+B·a=| a|^2 nbsp; nbsp; nbsp; (∵A·a=1) ∴B·a=| a|^2 1 ⋯(i) Also A×B=b nbsp; ...

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Byju's Answer

Standard XII

Physics

Vectors and Its Types

If vectors A ...

Question

If 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, then which of the following is/are true

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

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

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

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

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Solution

The correct option is C $\to A = \frac{(\to a \times \to b) + \to a}{| a |^{2}}$
We have $\to A + \to B = \to a$
Taking dot product of $\to a$ both sides, we get
$\to A \cdot \to a + \to B \cdot \to a = \to a \cdot \to a$
$\Rightarrow 1 + \to B \cdot \to a = | a |^{2}$
$(∵ \to A \cdot \to a = 1)$
$∴ \to B \cdot \to a = | a |^{2} - 1 \dots (i)$
Also $\to A \times \to B = \to b$
Taking cross product of $\to a$ both sides, we get
$\to a \times (\to A \times \to B) = \to a \times \to b$
$\Rightarrow (\to a \cdot \to B) \to A - (\to a \cdot \to A) \to B = \to a \times \to b$
$\Rightarrow (| a |^{2} - 1) \to A - \to B = \to a \times \to b \dots (i i)$ $(using (i) and \to a \cdot \to A = 1)$

Put $\to B = \to a - \to A$ in $(i i),$ we get
$\to A = \frac{(\to a \times \to b) + \to a}{| a |^{2}}$ $\Rightarrow \to B = \to a - ⎧ ⎪ ⎨ ⎪ ⎩ \frac{(\to a \times \to b) + \to a}{| a |^{2}} ⎫ ⎪ ⎬ ⎪ ⎭ = \frac{(\to b \times \to a) + \to a (| a |^{2} - 1)}{| a |^{2}}$

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If 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, then which of the following is/are true

If 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, then which of the following is/are true