Subtracting Vectors
Subtraction is addition with a flipped arrow. The vector
-\vec{v} is just \vec{v} turned around to
point the opposite way, so we define
\vec{u} - \vec{v} = \vec{u} + (-\vec{v}).
In components, subtract the matching entries:
\begin{bmatrix} u_x \\ u_y \end{bmatrix} - \begin{bmatrix} v_x \\ v_y \end{bmatrix} = \begin{bmatrix} u_x - v_x \\ u_y - v_y \end{bmatrix}.
The arrow from one tip to the other
There's a lovely picture for the difference. Draw \vec{u} and
\vec{v} from the same origin. Then
\vec{u} - \vec{v} is the arrow that runs from the tip of
\vec{v} to the tip of \vec{u} —
the displacement that takes you from \vec{v} to
\vec{u}. (Memory hook: it points to the first one named.)
Why this matters
The difference of two position vectors is the displacement between the two points — the single
most common thing you'll ever do with vectors. "How do I get from
A to B?" is answered by
\vec{B} - \vec{A}. In
machine learning
the same difference measures the error between a prediction and a target, and its
length is the gap you try to shrink.