The acute angle between the planes P₁ and P₂, when P₁ and P₂ are the planes passing through the intersection of the planes $5x + 8y + 13z - 29 = 0$ and $8x - 7y + z - 20 = 0$ and the points (2, 1, 3) and (0, 1, 2), respectively, is :

<p>Family of Plane's equation can be given by</p> <p>$(5 + 8\lambda )x + (8 - 7\lambda )y + (13 + \lambda )z - (29 + 20\lambda ) = 0$</p> <p>P₁ passes through (2, 1, 3)</p> <p>$$ \Rightarrow (10 + 16\lambda ) + (8 - 7\lambda ) + (39 + 3\lambda ) - (29 + 20\lambda ) = 0$$</p> <p>$\Rightarrow - 8\lambda + 28 = 0 \Rightarrow \lambda = {7 \over 2}$</p> <p>d.r, s of normal to P₁</p> <p>$\left\langle {33,{{ - 33} \over 2},{{33} \over 2}} \right\rangle$ or $\left\langle {1, - {1 \over 2},{1 \over 2}} \right\rangle$</p> <p>P₂ passes through (0, 1, 2)</p> <p>$\Rightarrow 8 - 7\lambda + 26 + 2\lambda - (29 + 20\lambda ) = 0$</p> <p>$\Rightarrow 5 - 25\lambda = 0$</p> <p>$\Rightarrow \lambda = {1 \over 5}$</p> <p>d.r, s of normal to P₂</p> <p>$\left\langle {{{33} \over 5},{{33} \over 5},{{66} \over 5}} \right\rangle$ or $\left\langle {1,1,2} \right\rangle$</p> <p>Angle between normals</p> <p>$$ = {{\left( {\widehat i - {1 \over 2}\widehat j + {1 \over 2}\widehat k} \right).\,\left( {\widehat i + \widehat j + 2\widehat k} \right)} \over {{{\sqrt 3 } \over 2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sqrt 6 }}$$</p> <p>$\cos \theta = {{1 - {1 \over 2} + 1} \over 3} = {1 \over 2}$</p> <p>$\theta = {\pi \over 3}$</p>