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# Determine whether the given planes are parallel or perpendicular, and in case they are neither, then find the angle between them. $(b)\; 2x + y + 3z - 2 = 0 \: and\: x -2y + 5 = 0$

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A)
• The direction ratios of normal to the plane $l_1=a_1x+b_1y+c_1 x=0$ are $a_1,b_1$ and $c_1$ and $l_2=a_2x+b_2y+c_2z=0$ are $a_2,b_2,c_2$
• If $L_1\perp L_2$,$a_1a_2+b_1b_2+c_1c_2=0$
The equation of the given planes are $2x+y+3z-2=0$ and $x-2y+5=0$
The direction cosines of $L_1$ are $(2,1,3)$
The direction cosines of $L_2$ are $(1,-2,5)$
$a_1a_2+b_1b_2+c_1c_2$ is $2\times 1+1\times -2+3\times 0$
$\Rightarrow 0$
$a_1a_2+b_1b_2+c_1c_2=0$