A Taste of Geometry Problem 4) State the geometric object defined by the followi

Question

A Taste of Geometry Problem 4)

State the geometric object defined by the following equation: x 2 + y 2 + z 2 2x 3y + 4z = 0

State any relevant information concerning this object. If you get this right, there will be two numbers that count as “relevant information”.

Computations With Vectors

Note for Problem 4 In the plane, R 2 , we defined the coordinate basis vectors: i = <1, 0> and j = <0, 1>. Any vector in R 2 can be defined in terms of i and j. In R 3 we have three such coordinate basis vectors: i = <1, 0, 0>, j = <0, 1, 0> and k = <0, 0, 1>.

Explanation / Answer

x 2 + y 2 + z 2 2x 3y + 4z = 0 is a sphere since

x y z have degree 2 with same coefficient

The sphere has centre at (1,3/2, -2)

Since sphere is an extension of circle in 3 Dimension

we have 3 coordinates for x,y,z as centre

Radius can be calculated by completion of squares

(x-1)2-1 +(y-3/2)2-9/4+(z+2)2-4 =0

Or (x-1)2 +(y-3/2)2+(z+2)2 = 29/4

Hence radius = (rt 29)/2

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