\overline{\mathrm{c}}$$ is $$\overline{\mathrm{r}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})=\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})$$

Plane $$P_1$$ is passing through the origin and parallel to vectors $$\overline{b_1}=2 \hat{j}+3 \hat{k}$$ and $$\overline{c_1}=4 \hat{j}-3 \hat{k}$$

$$\therefore \quad \overline{b_1} \times \overline{c_1}=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ 0 & 2 & 3 \\ 0 & 4 & -3 \end{array}\right|=-18 \hat{i}$$

$$\therefore \quad$$ Equation of $$P_1$$ is: $$r \cdot(-18 \mathrm{i})=0$$

Plane $$P_2$$ is passing through the origin and parallel to vectors $$\overline{b_2}=\hat{j}-\hat{k}$$ and $$\overline{c_2}=3 \hat{i}+3 \hat{j}$$

$$\therefore \quad \overline{b_2} \times \overline{c_2}=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ 0 & 1 & -1 \\ 3 & 3 & 0 \end{array}\right|=3 \hat{i}-3 \hat{j}-3 \hat{k}$$

$$\therefore \quad$$ Equation of $$P_2$$ is : $$r_2 \cdot(3 \hat{i}-3 \hat{j}-3 \hat{k})=0$$

Note that $$\overline{\mathrm{A}}$$ is parallel to the cross product of $$-18 \hat{i}$$ and $$3 \hat{i}-3 \hat{j}-3 \hat{k}$$

$$\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ -18 & 0 & 0 \\ 3 & -3 & -3 \end{array}\right|=-54 \hat{j}+54 \hat{k}$$

Let $$\theta$$ be the required angle.

$$\therefore \quad \theta=$$ Angle between $$54(-\hat{j}+\hat{k})$$ and $$2 \hat{i}+\hat{j}-2 \hat{k}$$

$$\begin{aligned} \therefore \quad \cos \theta & =\frac{54 \times(-1-2)}{54 \sqrt{0+1+1} \sqrt{4+1+4}} \\ & = \pm \frac{3}{3 \sqrt{2}} \\ & = \pm \frac{1}{\sqrt{2}} \\ \therefore \quad \theta=\frac{\pi}{4} & , \frac{3 \pi}{4} \end{aligned}$$

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