Browse Questions

# Find the centre and radius of the following spheres : $|\overrightarrow{2r}+(\overrightarrow{3i}-\overrightarrow{j}+\overrightarrow{4k})|=4$

Toolbox:
• Vector equation of a sphere with centre $\overrightarrow c$ and radius $a$ is $|\overrightarrow r-\overrightarrow c|=a$ When the centre is at the origin, the vector equation is $|\overrightarrow r|=a$ Cartesian equation is $(x-c_1)^2+(y-c_2)^2+(z-c_3)^2=a^2$ where $(c_1, c_2, c_3 )$ is the centre.
$|2\overrightarrow r+(3\overrightarrow i-\overrightarrow j+4\overrightarrow k)|=4 \Rightarrow 2|\overrightarrow r-(-\large\frac{3}{2}\overrightarrow i+\large\frac{1}{2}\overrightarrow j-2\overrightarrow k)|$ or
$| \overrightarrow r-(-\large\frac{3}{2}\overrightarrow i+\large\frac{1}{2}\overrightarrow j-2\overrightarrow k|=2$ pv of centre $\overrightarrow c=-\large\frac{3}{2}\overrightarrow i+\large\frac{1}{2}\overrightarrow j-2\overrightarrow k$
$\therefore c\bigg( -\large\frac{3}{2}, \large\frac{1}{2}, -2 \bigg)\: and \: r=2$