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# Determine whether the given planes are parallel or perpendicular, and in case they are neither ,then find the angles between them. $(d)\; 2x -y + 3z -1 = 0 \: and\: 2x -y + 3z + 3 = 0$

This question has multiple parts. Therefore each part has been answered as a separate question on Clay6.com

Toolbox:
• 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 $L_1\parallel L_2$ then $\large\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
Step 1:
The equation of the given planes are $2x-y+3z-1=0$ and $2x-y+3z+3=0$
The direction cosines of $L_1$ are $(2,-1,3)$
The direction cosines of $L_1$ are $(2,-1,3)$
Step 2:
Hence $\large\frac{a_1}{a_2}=\frac{2}{2}=$$1 \large\frac{b_1}{b_2}=\frac{-1}{-1}$$=1$
$\large\frac{c_1}{c_2}=\frac{3}{3}$$=1$
Therefore $\large\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
Therefore two planes are parallel to each other.