# The angle between the planes $2x-y+z=6\: and\: x+y +2z=7 \: is$ Option 1) $30^{\circ}$ Option 2) $45^{\circ}$ Option 3) $0^{\circ}$ Option 4) $60^{\circ}$

As we learnt in

Angle between two planes (Cartesian form) -

Let the two planes be

$ax+by+cz+d=0\: and \: a_{1}x+b_{1}y+c_{1}z+d_{1}=0$

then the angle between them is defined as the angle between their normals

$\cos \Theta = \frac{a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}$

-

$\cos \Theta = \frac{2\times 1+ (-1)X1+1\times2}{\sqrt{4+1+1 \ }\sqrt{1+1+4}}$

$\frac{3}{6}=\frac{1}{2}$

$\Theta = 60^{\circ}$

Option 1)

$30^{\circ}$

Incorrect

Option 2)

$45^{\circ}$

Incorrect

Option 3)

$0^{\circ}$

Incorrect

Option 4)

$60^{\circ}$

correct

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