The vector a and b are not perpendicular and c and d are two vectors satisfying: b × c = b × d and a ⋅ d = 0 . Then the vector d is equal to

b × c = b × d ⇒ a × ( b × c ) = a × ( b × d ) ⇒ ( a ⋅ c ) b − ( a ⋅ b ) c = ( a ⋅ d ) b − ( a ⋅ b ) d ⇒ ( a ⋅ c ) b − ( a ⋅ b ) c = − ( a ⋅ b ) d ∴ d = c − ( a b a ⋅ c ˉ ) b

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MathematicsVectorDot Product & Its Application ( Projection etc.)Hard2 minPYQ_2011

MathematicsHardsingle choice

The vector $a$ and $b$ are not perpendicular and $c$ and $d$ are two vectors satisfying: $b \times c = b \times d$ and $a \cdot d = 0$ . Then the vector $d$ is equal to

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Solution:

$\overline{b} \times \overline{c} = \overline{b} \times \overline{d}$ $\Rightarrow \overline{a} \times (\overline{b} \times \overline{c}) = \overline{a} \times (\overline{b} \times \overline{d})$ $\Rightarrow (\overline{a} \cdot \overline{c}) \overline{b} - (\overline{a} \cdot \overline{b}) \overline{c} = (\overline{a} \cdot \overline{d}) \overline{b} - (\overline{a} \cdot \overline{b}) \overline{d}$ $\Rightarrow (\overline{a} \cdot \overline{c}) \overline{b} - (\overline{a} \cdot \overline{b}) \overline{c} = - (\overline{a} \cdot \overline{b}) \overline{d}$ $∴ \overline{d} = \overline{c} - (\frac{a \cdot c ˉ}{a b}) \overline{b}$

Stream:JEESubject:MathematicsTopic:VectorSubtopic:Dot Product & Its Application ( Projection etc.)

⏱ 2mℹ️ Source: PYQ_2011

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