The acute angle between the planes P_{1} and | vedclass

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(A) The equation of any plane passing through the intersection of the planes $5x + 8y + 13z - 29 = 0$ and $8x - 7y + z - 20 = 0$ is given by $(5x + 8y

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$\frac{\pi}{3}$

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(A) The equation of any plane passing through the intersection of the planes $5x + 8y + 13z - 29 = 0$ and $8x - 7y + z - 20 = 0$ is given by $(5x + 8y + 13z - 29) + \lambda(8x - 7y + z - 20) = 0$.For plane $P_{1}$ passing through $(2, 1, 3)$:$(5(2) + 8(1) + 13(3) - 29) + \lambda(8(2) - 7(1) + 3 - 20) = 0$$(10 + 8 + 39 - 29) + \lambda(16 - 7 + 3 - 20) = 0$$28 + \lambda(-8) = 0 \Rightarrow \lambda = \frac{28}{8} = \frac{7}{2}$.Substituting $\lambda = \frac{7}{2}$ into the equation: $(5x + 8y + 13z - 29) + \frac{7}{2}(8x - 7y + z - 20) = 0 \Rightarrow 10x + 16y + 26z - 58 + 56x - 49y + 7z - 140 = 0 \Rightarrow 66x - 33y + 33z - 198 = 0 \Rightarrow 2x - y + z = 6$.The normal vector is $\vec{n_{1}} = 2\hat{i} - \hat{j} + \hat{k}$.For plane $P_{2}$ passing through $(0, 1, 2)$:$(5(0) + 8(1) + 13(2) - 29) + \lambda(8(0) - 7(1) + 2 - 20) = 0$$(8 + 26 - 29) + \lambda(-7 + 2 - 20) = 0$$5 + \lambda(-25) = 0 \Rightarrow \lambda = \frac{5}{25} = \frac{1}{5}$.Substituting $\lambda = \frac{1}{5}$ into the equation: $(5x + 8y + 13z - 29) + \frac{1}{5}(8x - 7y + z - 20) = 0 \Rightarrow 25x + 40y + 65z - 145 + 8x - 7y + z - 20 = 0 \Rightarrow 33x + 33y + 66z - 165 = 0 \Rightarrow x + y + 2z = 5$.The normal vector is $\vec{n_{2}} = \hat{i} + \hat{j} + 2\hat{k}$.The angle $	heta$ between the planes is given by $\cos 	heta = \frac{|\vec{n_{1}} \cdot \vec{n_{2}}|}{||\vec{n_{1}}|| ||\vec{n_{2}}||} = \frac{|(2)(1) + (-1)(1) + (1)(2)|}{\sqrt{2^{2} + (-1)^{2} + 1^{2}} \sqrt{1^{2} + 1^{2} + 2^{2}}} = \frac{|2 - 1 + 2|}{\sqrt{6} \sqrt{6}} = \frac{3}{6} = \frac{1}{2}$.Thus,$	heta = \cos^{-1}(\frac{1}{2}) = \frac{\pi}{3}$.

The acute angle between the planes $P_{1}$ and $P_{2}$,when $P_{1}$ and $P_{2}$ 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

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