Let a unit vector $\widehat{O P}$ make angles $\alpha, \beta, \gamma$ with the positive directions of the co-ordinate axes $\mathrm{OX}$, $\mathrm{OY}, \mathrm{OZ}$ respectively, where $\beta \in\left(0, \frac{\pi}{2}\right)$. If $\widehat{\mathrm{OP}}$ is perpendicular to the plane through points $(1,2,3),(2,3,4)$ and $(1,5,7)$, then which one of the following is true?
Solution
<p>Let $A \equiv (1,2,3),B \equiv (2,3,4),C \equiv (1,5,7)$</p>
<p>$$\overrightarrow n = \overrightarrow {AB} \times \overrightarrow {AC} = \left| {\matrix{
i & j & k \cr
1 & 1 & 1 \cr
0 & 3 & 4 \cr
} } \right|$$</p>
<p>$= \widehat i - 4\widehat j + 3\widehat k$</p>
<p>$$\widehat {OP} = {{ \pm (\widehat i - 4\widehat j + 3\widehat k)} \over {\sqrt {26} }}$$</p>
<p>Since $\cos \beta > 0$, take $-$ sign</p>
<p>$\widehat {OP} = {{\widehat i - 4\widehat j + 3\widehat k} \over {\sqrt {26} }}$</p>
<p>$\Rightarrow \cos \alpha < 0,\cos \gamma < 0$</p>
<p>$\alpha ,\gamma \in \left( {{\pi \over 2},\pi } \right)$</p>
About this question
Subject: Mathematics · Chapter: Vector Algebra · Topic: Types of Vectors
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