Vectors in a figure: routes in terms of given vectors

Vectors in a figure

To write a vector in a figure in terms of given vectors, travel from its starting point to its ending point along sides whose vectors you know. Add the vector of each side you travel along; a side travelled against its arrow counts with a minus sign.

\(KLMN\) is a parallelogram with \(\overrightarrow{KL}=\mathbf{p}\) and \(\overrightarrow{KN}=\mathbf{q}\), and \(KL\) is extended to \(T\) so that \(LT=KL\). Equal vectors: \(\overrightarrow{NM}=\mathbf{p}\), \(\overrightarrow{LM}=\mathbf{q}\) and \(\overrightarrow{LT}=\mathbf{p}\).

Method

1. Mark where the required vector starts and where it ends.
2. Choose a route between them along sides whose vectors are known.
3. Add the vectors along the route, with a minus sign for each side travelled against its arrow.
4. Simplify, and if possible check with a second route.

For \(\overrightarrow{TN}\): \(\overrightarrow{TN}=\overrightarrow{TK}+\overrightarrow{KN}=-2\mathbf{p}+\mathbf{q}\), because \(\overrightarrow{KT}=2\mathbf{p}\). Check by a second route: \(\overrightarrow{TN}=\overrightarrow{TL}+\overrightarrow{LM}+\overrightarrow{MN}=-\mathbf{p}+\mathbf{q}-\mathbf{p}=\mathbf{q}-2\mathbf{p}\).

Also \(\overrightarrow{KM}=\overrightarrow{KL}+\overrightarrow{LM}=\mathbf{p}+\mathbf{q}\) and \(\overrightarrow{NL}=\overrightarrow{NK}+\overrightarrow{KL}=\mathbf{p}-\mathbf{q}\).

Double arrows in a triangle

When a triangle shows two given vectors and a third side marked with a double arrow: if the two given vectors join head to tail, the double-arrowed side is their sum; if they start at the same point, it is a difference, "end minus start". If they end at the same point, it is also a difference: travel along one and against the other, for example \(\mathbf{a}-\mathbf{b}\). If the double arrow points the other way, the answer is the negative.

Check

Read the direction of every arrow before choosing signs: \(\overrightarrow{NL}=\mathbf{p}-\mathbf{q}\), but \(\overrightarrow{LN}=\mathbf{q}-\mathbf{p}\).