A vector $$\overrightarrow a = \alpha \widehat i + 2\widehat j + \beta \widehat k\left( {\alpha ,\beta \in R} \right)$$ lies in the plane of the vectors, $\overrightarrow b = \widehat i + \widehat j$ and $\overrightarrow c = \widehat i - \widehat j + 4\widehat k$. If $\overrightarrow a$ bisects the angle between $\overrightarrow b$ and $\overrightarrow c$, then:
Solution
Angle bisector $\overrightarrow a = \lambda \left( {\widehat b + \widehat c} \right)$
<br><br>= $$\lambda \left( {{{\widehat i + \widehat j} \over {\sqrt 2 }} + {{\widehat i - \widehat j + 4\widehat k} \over {3\sqrt 2 }}} \right)$$
<br><br>$\Rightarrow$ $$\overrightarrow a = {\lambda \over {3\sqrt 2 }}\left( {4\widehat i + 2\widehat j + 4\widehat k} \right)$$
<br><br>comparing with $\overrightarrow a = \alpha \widehat i + 2\widehat j + \beta \widehat k$
<br><br>${{2\lambda } \over {3\sqrt 2 }}$ = 2
<br><br>$\Rightarrow$ $\lambda$ = ${3\sqrt 2 }$
<br><br>$\therefore$ $\overrightarrow a = \left( {4\widehat i + 2\widehat j + 4\widehat k} \right)$
<br><br>Then $\overrightarrow a .\widehat k - 4$
<br><br>= 4 - 4 = 0
About this question
Subject: Mathematics · Chapter: Vector Algebra · Topic: Types of Vectors
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