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(D) The equations of the planes are $\vec{r} \cdot (\hat{i} + 2\hat{j} + 2\hat{k}) - 19 = 0$ and $\vec{r} \cdot (4\hat{i} - 3\hat{j} + 12\hat{k}) + 3

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$\vec{r} \cdot (25\hat{i} + 17\hat{j} + 62\hat{k}) = 238$

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(D) The equations of the planes are $\vec{r} \cdot (\hat{i} + 2\hat{j} + 2\hat{k}) - 19 = 0$ and $\vec{r} \cdot (4\hat{i} - 3\hat{j} + 12\hat{k}) + 3 = 0$.The equation of the angle bisector planes is given by:$\frac{\vec{r} \cdot (\hat{i} + 2\hat{j} + 2\hat{k}) - 19}{\sqrt{1^2 + 2^2 + 2^2}} = \pm \frac{\vec{r} \cdot (4\hat{i} - 3\hat{j} + 12\hat{k}) + 3}{\sqrt{4^2 + (-3)^2 + 12^2}}$$\frac{\vec{r} \cdot (\hat{i} + 2\hat{j} + 2\hat{k}) - 19}{3} = \pm \frac{\vec{r} \cdot (4\hat{i} - 3\hat{j} + 12\hat{k}) + 3}{13}$Case $1$ (Positive sign):$13(\vec{r} \cdot (\hat{i} + 2\hat{j} + 2\hat{k}) - 19) = 3(\vec{r} \cdot (4\hat{i} - 3\hat{j} + 12\hat{k}) + 3)$$\vec{r} \cdot (13\hat{i} + 26\hat{j} + 26\hat{k} - 12\hat{i} + 9\hat{j} - 36\hat{k}) = 9 + 247$$\vec{r} \cdot (\hat{i} + 35\hat{j} - 10\hat{k}) = 256$Case $2$ (Negative sign):$13(\vec{r} \cdot (\hat{i} + 2\hat{j} + 2\hat{k}) - 19) = -3(\vec{r} \cdot (4\hat{i} - 3\hat{j} + 12\hat{k}) + 3)$$\vec{r} \cdot (13\hat{i} + 26\hat{j} + 26\hat{k} + 12\hat{i} - 9\hat{j} + 36\hat{k}) = -9 + 247$$\vec{r} \cdot (25\hat{i} + 17\hat{j} + 62\hat{k}) = 238$Comparing with the options,the correct equation is $\vec{r} \cdot (25\hat{i} + 17\hat{j} + 62\hat{k}) = 238$.

Find the equation of the planes bisecting the angles between the planes $\vec{r} \cdot (\hat{i} + 2\hat{j} + 2\hat{k}) = 19$ and $\vec{r} \cdot (4\hat{i} - 3\hat{j} + 12\hat{k}) + 3 = 0$.

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