Adding and subtracting vectors Problem set B

Given \(\mathbf{p}=\begin{pmatrix}-5\\8\\1\end{pmatrix}\) and \(\mathbf{q}=\begin{pmatrix}3\\-2\\7\end{pmatrix}\), find:

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

Given \(\mathbf{s}=\begin{pmatrix}-8\\5\end{pmatrix}\) and \(\mathbf{t}=\begin{pmatrix}2\\-7\end{pmatrix}\), find the resultant of the two vectors and find the vector \(\mathbf{s}-\mathbf{t}\).

1. (a) The resultant \(\mathbf{s}+\mathbf{t}\)
2. (b) \(\mathbf{s}-\mathbf{t}\)

In the diagram, \(\overrightarrow{PQ}=\mathbf{x}\), \(\overrightarrow{QR}=\mathbf{y}\) and \(\overrightarrow{RS}=\mathbf{z}\). Write each of these in terms of \(\mathbf{x}\), \(\mathbf{y}\) and \(\mathbf{z}\).

1. (a) \(\overrightarrow{PR}\)
2. (b) \(\overrightarrow{QS}\)
3. (c) \(\overrightarrow{PS}\)

In the diagram, \(\overrightarrow{PQ}=\mathbf{x}\), \(\overrightarrow{QR}=\mathbf{y}\) and \(\overrightarrow{RS}=\mathbf{z}\). Write each of these in terms of \(\mathbf{x}\), \(\mathbf{y}\) and \(\mathbf{z}\).

1. (a) \(\overrightarrow{SP}\)
2. (b) \(\overrightarrow{RP}\)
3. (c) \(\overrightarrow{SQ}\)

A helicopter flies from a base H to a village J, a displacement of \(\begin{pmatrix}-4\\9\end{pmatrix}\) km, and then from J to a hospital K, a displacement of \(\begin{pmatrix}10\\-6\end{pmatrix}\) km. The first component is east and the second is north.

1. (a) Find the vector \(\overrightarrow{HK}\) from H to K.
2. (b) Find the vector \(\overrightarrow{KH}\) from K back to H.

From a lighthouse L, the displacement to a buoy M is \(\begin{pmatrix}-3\\6\end{pmatrix}\) km and the displacement to a wreck N is \(\begin{pmatrix}8\\-2\end{pmatrix}\) km (east, north).

1. (a) Find the vector \(\overrightarrow{MN}\) from M to N.
2. (b) Find the vector \(\overrightarrow{NM}\) from N to M.

In the diagram, \(\overrightarrow{PQ}=\mathbf{x}\), \(\overrightarrow{QR}=\mathbf{y}\) and \(\overrightarrow{SR}=\mathbf{z}\). Write each of these in terms of \(\mathbf{x}\), \(\mathbf{y}\) and \(\mathbf{z}\).

1. (a) \(\overrightarrow{PS}\)
2. (b) \(\overrightarrow{QS}\)
3. (c) \(\overrightarrow{SP}\)

In the quadrilateral WXYZ, \(\overrightarrow{WX}=\begin{pmatrix}-1\\6\\3\end{pmatrix}\), \(\overrightarrow{XY}=\begin{pmatrix}5\\-4\\0\end{pmatrix}\) and \(\overrightarrow{YZ}=\begin{pmatrix}-2\\1\\7\end{pmatrix}\). Find the column vectors \(\overrightarrow{WZ}\) and \(\overrightarrow{ZX}\).

1. (a) \(\overrightarrow{WZ}\)
2. (b) \(\overrightarrow{ZX}\)