Write as a Vector Equality x^2+y^2=1 , x^2+(y+2)^2=9

Problem

x^2+y^2=1,x^2+(y+2)^2=9

Solution

1. Identify the standard form of the equations as circles in the Cartesian plane.

2. Define the position vector r as the vector from the origin to a point (x,y) on the curve.

r=([x],[y])

3. Represent the first equation x^2+y^2=1 using the dot product of the vector with itself.

r⋅r=1

4. Represent the second equation x^2+(y+2)^2=9 by defining a center vector c

c=([0],[−2])

5. Apply the vector magnitude formula for the distance between the point r and the center c

|r−c|^2=9

6. Combine the expressions into a single vector equality statement.

Final Answer

r⋅r=1,|r−([0],[−2])|^2=9

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