If A→+B→=R→ and 2A→+B→ is perpendicular to B→ then

A→+B→=R→given, 2A→+B→ is perpendicular to B→.2A→+B→.B→=0⇒A→+A→+B→.B→=0 A→+R→.B→=0⇒A→=-R→ ∴A=R

If A→+B→=R→ and 2A→+B→ is perpendicular to B→ then

A→+B→=R→given, 2A→+B→ is perpendicular to B→.2A→+B→.B→=0⇒A→+A→+B→.B→=0 A→+R→.B→=0⇒A→=-R→ ∴A=R

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If $\vec{A} + \vec{B} = \vec{R}$ and $2 \vec{A} + \vec{B}$ is perpendicular to $\vec{B}$ then

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A = R

B = 2R

B = R

B = A

answer is A.

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Detailed Solution

$\vec{A} + \vec{B} = \vec{R}$
given, $2 \vec{A} + \vec{B}$ is perpendicular to $\vec{B}$ .
$(2 \vec{A} + \vec{B}) . \vec{B} = 0 \Rightarrow [\vec{A} + (\vec{A} + \vec{B})] . \vec{B} = 0 (\vec{A} + \vec{R}) . \vec{B} = 0 \Rightarrow \vec{A} = - \vec{R} ∴ A = R$