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Question
The vector equation of the plane passing through \[\vec{a} , \vec{b} , \vec{c} ,\text{ is }\vec{r} = \alpha \vec{a} + \beta \vec{b} + \gamma \vec{c} ,\] provided that
Options
α + β + γ = 0
α + β + γ =1
α + β = γ
α2 + β2 + γ2 = 1
Solution
α + β + γ =1
Given: A plane passing through \[\vec{a} , \vec{b} , \vec{c}\]
⇒ Lines \[\vec{a} - \vec{b}\] and \[\vec{c} - \vec{a}\] lie on the plane.
The parmetric equation of the plane can be written as:
\[\begin{array}{l}\vec{r} = \vec{a} + \lambda_1 ( \vec{a} - \vec{b} ) + \lambda_2 ( \vec{c} - \vec{a} ) \\ \vec{r} = \vec{a} (1 + \lambda_1 - \lambda_2 ) - \lambda_1 \vec{b} + \lambda_2 \\ \text{ Given that }\vec{r} = \alpha \vec{a} + \beta \vec{b} + \gamma \vec{c} \\ \therefore \alpha + \beta + \gamma = 1 + \lambda_1 - \lambda_2 -1\end{array}\]
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