Addition of Vectors

Motivating the definition. Imagine a unit mass at the origin, pushed by two unit forces $\vec a$ (along the positive $x$-axis) and $\vec b$ (along the positive $y$-axis), acting one after the other: first $\vec a$ moves the object from $(0,0)$ to $(1,0)$, then $\vec b$ moves it from $(1,0)$ to $(1,1)$. The net displacement is the single segment from $(0,0)$ to $(1,1)$ — and this is exactly what we mean by $\vec a+\vec b$. The same idea works even when the two vectors are not perpendicular and not of equal magnitude: bring the tail of the second vector to the tip of the first, and the sum is the vector from the very first tail to the very last tip.

Triangle law of addition. If two vectors are represented, in magnitude and direction, by two sides of a triangle taken in order (tail-to-tip), then their sum is represented by the third side of the triangle, taken in the reverse order (from the start of the first vector to the end of the second).

Result. If $\vec a,\vec b,\vec c$ are the three sides of a triangle taken in order (so each one starts where the previous one ended, and the last one returns to the start), then $\vec a+\vec b+\vec c=\vec 0.$ Proof. Write $\vec a=\vec{AB},\ \vec b=\vec{BC},\ \vec c=\vec{CA}$. Then $\vec a+\vec b+\vec c=\vec{AB}+\vec{BC}+\vec{CA}=\vec{AC}+\vec{CA}=\vec{AA}=\vec 0$. …