Question

Find the vector equation of the sphere concentric with the sphere $|\overrightarrow{r}+(\hat{i}-2\hat{j}-3\hat{k})|=5$ and its radius is two times of that sphere.

Answer 1

Vector equation of sphere
$|\overrightarrow{r}+(\hat{i}-2\hat{j}-3\hat{k})|=5$
Centre of sphere
$\overrightarrow{c}=(-\hat{i}+2\hat{j}+3\hat{k})$
and $a=5$
Double radius $2a=2\times 5=10$
Hence required vector equation be
$|\overrightarrow{r}-\left(\overrightarrow{c}\right)|=a$
$|\overrightarrow{r}-(-\hat{i}+2\hat{j}+3\hat{k})|=10$
$|\overrightarrow{r}+(\hat{i}-2\hat{j}-3\hat{k})|=10$.