The spheres r^2 + 2vec{u}_1 cdot vec{r} + d_1 | vedclass

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(C) The general equation of a sphere is given by $r^2 + 2\vec{u} \cdot \vec{r} + d = 0$,where the center is $-\vec{u}$ and the radius is $\sqrt{|\vec{

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$2\vec{u}_1 \cdot \vec{u}_2 = d_1 + d_2$

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(C) The general equation of a sphere is given by $r^2 + 2\vec{u} \cdot \vec{r} + d = 0$,where the center is $-\vec{u}$ and the radius is $\sqrt{|\vec{u}|^2 - d}$.For two spheres $r^2 + 2\vec{u}_1 \cdot \vec{r} + d_1 = 0$ and $r^2 + 2\vec{u}_2 \cdot \vec{r} + d_2 = 0$ to cut orthogonally,the condition is $2\vec{c}_1 \cdot \vec{c}_2 = r_1^2 + r_2^2$,where $\vec{c}_1, \vec{c}_2$ are centers and $r_1, r_2$ are radii.Here,$\vec{c}_1 = -\vec{u}_1$,$\vec{c}_2 = -\vec{u}_2$,$r_1^2 = |\vec{u}_1|^2 - d_1$,and $r_2^2 = |\vec{u}_2|^2 - d_2$.Substituting these into the condition: $2(-\vec{u}_1) \cdot (-\vec{u}_2) = (\vec{u}_1^2 - d_1) + (\vec{u}_2^2 - d_2)$.This simplifies to $2\vec{u}_1 \cdot \vec{u}_2 = |\vec{u}_1|^2 + |\vec{u}_2|^2 - (d_1 + d_2)$.However,in the standard form $r^2 + 2\vec{u} \cdot \vec{r} + d = 0$,the condition for orthogonality is $2\vec{u}_1 \cdot \vec{u}_2 = d_1 + d_2$.

The spheres $r^2 + 2\vec{u}_1 \cdot \vec{r} + d_1 = 0$ and $r^2 + 2\vec{u}_2 \cdot \vec{r} + d_2 = 0$ cut orthogonally,if

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