Why vector C is equal to vector A - B?

Question

I'm new into multivariable calculus and I can't help but ask myself, Why vector $\vec{C}$ is equal to $\vec{A} -\vec{B}$?

Edit: + corrected to -

Answer 1

If you stand in point A and walk $\vec{B}$ you end up in point C. Then you walk $\vec{C}$ and end up in B, the same place where you would have been, if you walked $\vec{A}$ in the beginning. So $$ \vec{B}+\vec{C}=\vec{A} \Leftrightarrow \vec{A}-\vec{B}=\vec{C} $$ Note that the starting point doesn't matter. You have (by looking at the graph) $$ \vec{A}=\begin{pmatrix}1\\2\end{pmatrix} $$ $$ \vec{B}=\begin{pmatrix}2\\-2\end{pmatrix} $$ $$ \vec{C}=\begin{pmatrix}-1\\4\end{pmatrix} $$ Now you can clearly see, that $$ \vec{A}-\vec{B}=\begin{pmatrix}1\\2\end{pmatrix} -\begin{pmatrix}2\\-2\end{pmatrix} =\begin{pmatrix}-1\\4\end{pmatrix} =\vec{C}\qquad $$

Answer 2

This is not calculus, just vector algebra.

We have $a=b+c$, by the definition of vector sum (parallelogram law).

By definition of subtraction, $c=a-b$ exactly when $a=b+c$.