If \[\vec{a} \cdot \vec{a} = 0 \text{ and } \vec{a} \cdot \vec{b} = 0,\] what can you conclude about the vector \[\vec{b}\] ?

\[\text{ Given that } \vec{a} . \vec{a} = 0\] \[ \Rightarrow \left| \vec{a} \right|^2 = 0\] \[ \Rightarrow \left| \vec{a} \right| = 0 . . . \left( 1 \right)\] \[\text{ Also given that }\] \[ \vec{a} . \vec{b} = 0\] \[ \Rightarrow \left| \vec{a} \right| \left| \vec{b} \right| \cos \theta = 0............ (\text{ Where } \theta \text{ is the angle between }\vec{a} \text{ and } \vec{b} )\] \[ \Rightarrow 0 \left| \vec{b} \right| \cos \theta = 0....................... [\text{ From } (1)]\] \[ \Rightarrow 0 = 0\] \[\text{ So, it means that for any vector } \vec{b} , \text{ the given equation }\vec{a} . \vec{b} = 0 \text{ is satisfied }.\]

If → a ⋅ → a = 0 and → a ⋅ → B = 0 , What Can You Conclude About the Vector → B ? - Mathematics

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If $\vec{a} \cdot \vec{a} = 0 and \vec{a} \cdot \vec{b} = 0,$ what can you conclude about the vector $\vec{b}$ ?

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उत्तर

$Given that \vec{a} . \vec{a} = 0$

$\Rightarrow {| \vec{a} |}^{2} = 0$

$\Rightarrow | \vec{a} | = 0 . . . (1)$

$Also given that$

$\vec{a} . \vec{b} = 0$

$\Rightarrow | \vec{a} | | \vec{b} | \cos θ = 0. . . . . . . . . . . . (Where θ is the angle between \vec{a} and \vec{b})$

$\Rightarrow 0 | \vec{b} | \cos θ = 0. . . . . . . . . . . . . . . . . . . . . . . [From (1)]$

$\Rightarrow 0 = 0$

$So, it means that for any vector \vec{b}, the given equation \vec{a} . \vec{b} = 0 is satisfied .$

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Multiplication of a Vector by a Scalar

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Q 39 Q 38 Q 40

अध्याय 24: Scalar Or Dot Product - Exercise 24.1 [पृष्ठ ३२]

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आरडी शर्मा Mathematics [English] Class 12

अध्याय 24 Scalar Or Dot Product
Exercise 24.1 | Q 39 | पृष्ठ ३२

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Multiplication of a Vector by a Scalar

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If → a ⋅ → a = 0 and → a ⋅ → B = 0 , What Can You Conclude About the Vector → B ? - Mathematics

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If → a ⋅ → a = 0 and → a ⋅ → B = 0 , What Can You Conclude About the Vector → B ? - Mathematics

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