1) Draw a picture of two spheres of different sizes being tangent to each other at exactly one point P in R 3 . Do NOT put one sphere inside the other one. 2) Suppose we have two surfaces S 1 and S 2...

1) Draw a picture of two spheres of different sizes being tangent to each other at exactly one point P in R
³
. Do NOT put one sphere inside the other one.

2) Suppose we have two surfaces S₁
and S₂
which intersect at some point P = (x₀, y₀, z₀), and further suppose that the normal vectors n₁
=‹a₁, b₁, c₁› (for S₁) and n₂
= ‹a₂, b₂, c₂›(for S₂) at P are parallel (and non-zero). Prove that the tangent planes T_PS₁
and T_PS₂
are the same by transforming the scalar equation for T_PS₁
into the scalar equation for T_PS₂. Hint: This should be a one-step algebraic transformation based upon using the algebraic definition of parallel vectors.