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  4. If a makes an acute angle with b , r ⋅ a =0 and r × b = c × b, then r =

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Q. If $a$ makes an acute angle with $b , r \cdot a =0$ and $r \times b = c \times b$, then $r =$

AP EAMCETAP EAMCET 2019

Solution:

Given,
and $ r \cdot a =0 $
$ r \times b = c \times b $
So, $ a \times( r \times b )= a \times( c \times b ) $
$\Rightarrow r ( a \cdot b )- b ( a \cdot r )= c ( a \cdot b )- b ( a \cdot c ) $
$ \Rightarrow r ( a \cdot b )= c ( a \cdot b )- b ( a \cdot c ) $
$ \Rightarrow r = c - b \left(\frac{ c \cdot a }{ a \cdot b }\right)$