Add the vector (1, 2)^{T} to
the __point__ (4, 4)^{T}.

The result is a point:
(4, 4)^{T} + (1, 2)^{T} = ( 5, 6 )^{T}

If the vector (1, 2)^{T}
is a displacement (i.e., an amount by which to change x, and an amount by
which to change y),
then the result must be a point in a new location:

(4, 4)^{T}+ (1, 2)^{T}= ( 5, 6 )^{T}

This is one of those situations where using the same representation (i.e. column matrices) for both points and vectors is confusing. You have to keep track of what type of object each matrix represents.

A displacement vector added to a point results in a point. Here is a diagram of that:

This is slightly odd.
Two mathematical objects *of different types*
are added together.
Again, the "+" sign is overloaded.

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