Mathematics

Problem

Find the equation of the sphere described on the joint of points A and B having position position vectors

2\hat{i} + 6\hat{j} - 7\hat{k}

and

-2\hat{i} + 4\hat{j} - 3\hat{k}

, respectively, as the diameter. Find the centre and the radius of the sphere.

Intuitive explanation

A sphere with a given diameter has a special property: any point on the sphere satisfies the condition that the angle subtended by the diameter endpoints is always a right angle. This means if $P$ is any point on the sphere, then $\vec{PA} \cdot \vec{PB} = 0$ . The center is simply the midpoint of the diameter, and the radius is half the distance between $A$ and $B$ —use these geometric facts to build the sphere's equation efficiently.

Intuitive explanation

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