Q.If $\vec a = 2\hat i+3\hat k$, $\vec b = \hat i+2\hat j-3\hat k$ and $\vec c = 3\hat j-4\hat k$, then determine $\vec a - \vec b + 2\vec c$.

Triangle Law of Vector Addition

How do you combine two vectors into a single one? If you make two journeys one after the other, the net journey is a single vector from where you started to where you finished. That is exactly the triangle law.

The law

Place $\vec{a}$, then start $\vec{b}$ where $\vec{a}$ ends. The arrow that closes the triangle, drawn from the start of $\vec{a}$ to the end of $\vec{b}$, is the resultant $\vec{a}+\vec{b}$.

Why it works

Read the vectors as directed displacements: going from $A$ to $B$ and then $B$ to $C$ lands you at $C$, and the single displacement that achieves the same is $A$ to $C$. The intermediate point $B$ cancels — only the overall start and finish survive.

Consequences

• Commutative: $\vec{a}+\vec{b}=\vec{b}+\vec{a}$. Completing the triangle the other way gives the same closing side — which is why the parallelogram law agrees with the triangle law.
• Closed triangle = zero: if three vectors form a triangle taken in order, $\vec{AB}+\vec{BC}+\vec{CA}=\vec{0}$, since you return to the start.
• To subtract, add the negative: $\vec{a}-\vec{b}=\vec{a}+(-\vec{b})$, reversing $\vec{b}$ before joining it. …