Write as a Vector Equality x^2+y^2=100 , y=-x-2

Problem

x2+y2=100,y=−x−2

Solution

1. Identify the components of a generic position vector r in two dimensions, which is typically represented as r=<x,y>

2. Substitute the given expression for y into the vector components to express the vector in terms of a single variable x

r=<x,−x−2>

3. Express the constraint x2+y2=100 using the dot product definition of the magnitude squared, where |r|2=r⋅r

<x,y>⋅<x,y>=100

4. Combine the substitution and the magnitude constraint to write the system as a single vector equality involving the norm.

|<x,−x−2>|2=100

Final Answer

r=<x,−x−2>,|r|2=100

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