The acute angle between the planes P_1 and P_2, when P_1 and P_2 are the planes passing th…

Question

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:

Accepted Answer

Family: $(5+8\lambda)x+(8-7\lambda)y+(13+\lambda)z-(29+20\lambda)=0$.

$P_1$ through $(2,1,3)$: $\lambda=\dfrac72$, giving normal $\propto\left(1,-\dfrac12,\dfrac12\right)$.

$P_2$ through $(0,1,2)$: $\lambda=\dfrac15$, giving normal $\propto(1,1,2)$.
$$\cos\theta=\frac{1-\frac12+1}{\sqrt{\frac32}\cdot\sqrt6}=\frac12 \implies \theta=\frac{\pi}{3}.$$

Solutions · 1