Multivariable and Vector Calculus · Geometry, vectors, and vector-valued functions
Let A=(1,-2,4) and B=(5,2,-2)
Problem
Let \(A=(1,-2,4)\) and \(B=(5,2,-2)\). (a) Find the exact distance \(|AB|\). (b) Find the midpoint \(M\) of the segment \(AB\). (c) Write an equation of the sphere that has the segment \(AB\) as a diameter. (d) Determine whether the point \(C=(3,4,1)\) lies on that sphere. Justify your conclusion by comparing \(|CM|\) with the radius. (e) Determine whether \(A\), \(B\), and \(C\) are collinear.
Hint
Distance and midpoint are read from the components of \(\overrightarrow{AB}\). A sphere with diameter \(AB\) is centered at the midpoint.
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