Adding and subtracting vectors Problem set A

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

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

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

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) \(\overrightarrow{AC}\)
2. (b) \(\overrightarrow{BD}\)
3. (c) \(\overrightarrow{AD}\)

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) \(\overrightarrow{DA}\)
2. (b) \(\overrightarrow{CA}\)
3. (c) \(\overrightarrow{DB}\)

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

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

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) \(\overrightarrow{AC}\)
2. (b) \(\overrightarrow{AD}\)
3. (c) \(\overrightarrow{DB}\)

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