Adding and subtracting vectors Problem set A Worked solutions

Given \(\mathbf{a}=\begin{pmatrix}9\\-3\\4\end{pmatrix}\) and \(\mathbf{b}=\begin{pmatrix}2\\6\\-5\end{pmatrix}\), find:

1. (a) \(\mathbf{a}-\mathbf{b}\)

   • \(\mathbf{a}-\mathbf{b}=\mathbf{a}+(-\mathbf{b})=\begin{pmatrix}9\\-3\\4\end{pmatrix}+\begin{pmatrix}-2\\-6\\5\end{pmatrix}\)
   • \(=\begin{pmatrix}7\\-9\\9\end{pmatrix}\)

2. (b) \(\mathbf{b}-\mathbf{a}\)

   • \(\mathbf{b}-\mathbf{a}=\begin{pmatrix}2\\6\\-5\end{pmatrix}+\begin{pmatrix}-9\\3\\-4\end{pmatrix}\)
   • \(=\begin{pmatrix}-7\\9\\-9\end{pmatrix}\)

Given \(\mathbf{u}=\begin{pmatrix}6\\-4\end{pmatrix}\) and \(\mathbf{v}=\begin{pmatrix}-3\\9\end{pmatrix}\), find the resultant of the two vectors and find the vector \(\mathbf{u}-\mathbf{v}\).

1. (a) The resultant \(\mathbf{u}+\mathbf{v}\)

   • Add the components, row by row
   • \(\mathbf{u}+\mathbf{v}=\begin{pmatrix}3\\5\end{pmatrix}\)

2. (b) \(\mathbf{u}-\mathbf{v}\)

   • Subtracting is adding the negative of the second vector
   • \(\mathbf{u}-\mathbf{v}=\begin{pmatrix}6\\-4\end{pmatrix}+\begin{pmatrix}3\\-9\end{pmatrix}\)
   • \(=\begin{pmatrix}9\\-13\end{pmatrix}\)

In the diagram, \(\overrightarrow{AB}=\mathbf{a}\), \(\overrightarrow{BC}=\mathbf{b}\) and \(\overrightarrow{CD}=\mathbf{c}\). Write each of these in terms of \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\).

1. (a) \(\overrightarrow{AC}\)

   • \(\overrightarrow{AC}=\overrightarrow{AB}+\overrightarrow{BC}\)
   • \(\overrightarrow{AC}=\mathbf{a}+\mathbf{b}\)

2. (b) \(\overrightarrow{BD}\)

   • \(\overrightarrow{BD}=\overrightarrow{BC}+\overrightarrow{CD}\)
   • \(\overrightarrow{BD}=\mathbf{b}+\mathbf{c}\)

3. (c) \(\overrightarrow{AD}\)

   • \(\overrightarrow{AD}=\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CD}\)
   • \(\overrightarrow{AD}=\mathbf{a}+\mathbf{b}+\mathbf{c}\)

In the diagram, \(\overrightarrow{AB}=\mathbf{a}\), \(\overrightarrow{BC}=\mathbf{b}\) and \(\overrightarrow{CD}=\mathbf{c}\). Write each of these in terms of \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\).

1. (a) \(\overrightarrow{DA}\)

   • \(\overrightarrow{AD}=\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CD}=\mathbf{a}+\mathbf{b}+\mathbf{c}\)
   • \(\overrightarrow{DA}=-\overrightarrow{AD}=-(\mathbf{a}+\mathbf{b}+\mathbf{c})=-\mathbf{a}-\mathbf{b}-\mathbf{c}\)

2. (b) \(\overrightarrow{CA}\)

   • \(\overrightarrow{AC}=\overrightarrow{AB}+\overrightarrow{BC}=\mathbf{a}+\mathbf{b}\)
   • \(\overrightarrow{CA}=-\overrightarrow{AC}=-(\mathbf{a}+\mathbf{b})=-\mathbf{a}-\mathbf{b}\)

3. (c) \(\overrightarrow{DB}\)

   • \(\overrightarrow{BD}=\overrightarrow{BC}+\overrightarrow{CD}=\mathbf{b}+\mathbf{c}\)
   • \(\overrightarrow{DB}=-\overrightarrow{BD}=-(\mathbf{b}+\mathbf{c})=-\mathbf{b}-\mathbf{c}\)

A ship sails from port P to an island Q, a displacement of \(\begin{pmatrix}12\\5\end{pmatrix}\) km, and then from Q to a second island R, a displacement of \(\begin{pmatrix}-3\\8\end{pmatrix}\) km. The first component is east and the second is north.

1. (a) Find the vector \(\overrightarrow{PR}\) from P to R.

   • \(\overrightarrow{PR}=\overrightarrow{PQ}+\overrightarrow{QR}=\begin{pmatrix}12\\5\end{pmatrix}+\begin{pmatrix}-3\\8\end{pmatrix}\)
   • \(=\begin{pmatrix}9\\13\end{pmatrix}\)

2. (b) Find the vector \(\overrightarrow{RP}\) from R back to P.

   • \(\overrightarrow{RP}=-\overrightarrow{PR}\)
   • \(=\begin{pmatrix}-9\\-13\end{pmatrix}\)

From a radio tower T, the displacement to a tree U is \(\begin{pmatrix}7\\3\end{pmatrix}\) km and the displacement to a rock V is \(\begin{pmatrix}2\\10\end{pmatrix}\) km (east, north).

1. (a) Find the vector \(\overrightarrow{UV}\) from U to V.

   • \(\overrightarrow{UV}=\overrightarrow{UT}+\overrightarrow{TV}=-\overrightarrow{TU}+\overrightarrow{TV}\)
   • \(=\begin{pmatrix}-7\\-3\end{pmatrix}+\begin{pmatrix}2\\10\end{pmatrix}\)
   • \(=\begin{pmatrix}-5\\7\end{pmatrix}\)

2. (b) Find the vector \(\overrightarrow{VU}\) from V to U.

   • \(\overrightarrow{VU}=-\overrightarrow{UV}\)
   • \(=\begin{pmatrix}5\\-7\end{pmatrix}\)

In the diagram, \(\overrightarrow{AB}=\mathbf{a}\), \(\overrightarrow{CB}=\mathbf{b}\) and \(\overrightarrow{CD}=\mathbf{c}\). Write each of these in terms of \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\).

1. (a) \(\overrightarrow{AC}\)

   • \(\overrightarrow{AC}=\overrightarrow{AB}-\overrightarrow{CB}\)
   • \(\overrightarrow{AC}=\mathbf{a}-\mathbf{b}\)

2. (b) \(\overrightarrow{AD}\)

   • \(\overrightarrow{AD}=\overrightarrow{AB}-\overrightarrow{CB}+\overrightarrow{CD}\)
   • \(\overrightarrow{AD}=\mathbf{a}-\mathbf{b}+\mathbf{c}\)

3. (c) \(\overrightarrow{DB}\)

   • \(\overrightarrow{DB}=-\overrightarrow{CD}+\overrightarrow{CB}\)
   • \(\overrightarrow{DB}=-\mathbf{c}+\mathbf{b}\)
   • \(\overrightarrow{DB}=\mathbf{b}-\mathbf{c}\)

In the quadrilateral ABCD, \(\overrightarrow{AB}=\begin{pmatrix}4\\-2\\1\end{pmatrix}\), \(\overrightarrow{BC}=\begin{pmatrix}-3\\5\\2\end{pmatrix}\) and \(\overrightarrow{CD}=\begin{pmatrix}2\\0\\-6\end{pmatrix}\). Find the column vectors \(\overrightarrow{AD}\) and \(\overrightarrow{DB}\).

1. (a) \(\overrightarrow{AD}\)

   • \(\overrightarrow{AD}=\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CD}\)
   • \(=\begin{pmatrix}4\\-2\\1\end{pmatrix}+\begin{pmatrix}-3\\5\\2\end{pmatrix}+\begin{pmatrix}2\\0\\-6\end{pmatrix}=\begin{pmatrix}3\\3\\-3\end{pmatrix}\)

2. (b) \(\overrightarrow{DB}\)

   • \(\overrightarrow{BD}=\overrightarrow{BC}+\overrightarrow{CD}=\begin{pmatrix}-1\\5\\-4\end{pmatrix}\)
   • \(\overrightarrow{DB}=-\overrightarrow{BD}=\begin{pmatrix}1\\-5\\4\end{pmatrix}\)