Equal vectors in a parallelogram on a grid

Equal vectors in a parallelogram

The name of a parallelogram lists its vertices in order round the shape. In parallelogram \(ABCD\) the sides run \(A\to B\to C\to D\to A\), so the opposite sides give two pairs of equal vectors: \(\overrightarrow{AB}=\overrightarrow{DC}\) and \(\overrightarrow{AD}=\overrightarrow{BC}\).

In the figure \(\overrightarrow{AB}=\overrightarrow{DC}=\begin{pmatrix}6\\ 2\end{pmatrix}\) (one arrowhead) and \(\overrightarrow{AD}=\overrightarrow{BC}=\begin{pmatrix}-1\\ 5\end{pmatrix}\) (two arrowheads). Read both vectors of a pair in the same direction: \(\overrightarrow{CD}\) runs the other way, so \(\overrightarrow{CD}=-\overrightarrow{AB}\).

Method: finding the fourth vertex

1. Write the three known vertices in the order given by the name.
2. Pick the side whose two ends are both known, for example \(\overrightarrow{AB}\) in \(ABCD\) when \(A\), \(B\) and \(D\) are known.
3. Move the vertex that starts the opposite side by the same vector: \(\overrightarrow{DC}=\overrightarrow{AB}\), so \(C\) is \(D\) moved by \(\overrightarrow{AB}\).
4. Check the other pair of sides: \(\overrightarrow{AD}\) should equal \(\overrightarrow{BC}\).

Example: \(A(3,\ 2)\), \(B(9,\ 4)\) and \(D(2,\ 7)\) give \(\overrightarrow{AB}=\begin{pmatrix}6\\ 2\end{pmatrix}\), so \(C\) is \((2+6,\ 7+2)=(8,\ 9)\). Check: \(\overrightarrow{AD}=\begin{pmatrix}-1\\ 5\end{pmatrix}\) and \(\overrightarrow{BC}=\begin{pmatrix}-1\\ 5\end{pmatrix}.\)

The name decides the shape

With the same three points, parallelogram \(ABDC\) runs \(A\to B\to D\to C\), so \(\overrightarrow{AB}=\overrightarrow{CD}\) and \(\overrightarrow{DC}=-\overrightarrow{AB}\), so the fourth vertex \(C\) is \(D\) moved by \(-\overrightarrow{AB}\), that is \((2-6,\ 7-2)=(-4,\ 5)\). Check: \(\overrightarrow{AC}=\begin{pmatrix}-7\\ 3\end{pmatrix}=\overrightarrow{BD}\).

Check

Opposite sides of a parallelogram always have equal magnitudes, but they are equal vectors only when both are read in the same direction. Two vectors can have the same magnitude and opposite directions, such as \(\overrightarrow{AB}\) and \(\overrightarrow{CD}\) in \(ABCD\).