Find  \(x\), \(y\) and \(z\)  if  \(\left( {\begin{array}{*{20}{c}}
{x – 3}\\
{y + 2}\\
{z + 4}
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
1\\
3\\
{ – 2}
\end{array}} \right)\)

Find scalars  \(a\), \(b\) and \(c\),   if  \(2\left( {\begin{array}{*{20}{c}}
1\\
2\\
{c + 1}
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
{a – 1}\\
b\\
6
\end{array}} \right)\).

Find \(k\),   correcting to two significant figures, such a vector  \(k\left( {\begin{array}{*{20}{c}}
2\\
{ – 3}\\
2
\end{array}} \right)\) is in the same direction as  \(\left( {\begin{array}{*{20}{c}}
2\\
{ – 3}\\
2
\end{array}} \right)\)  and with length 4 units.

Find   \(\vec a \bullet \vec b\)  if  \(\left| {\vec a} \right| = 2,\left| {\vec b} \right| = 4,\theta = 130^\circ \),   correcting to two significant figures.

Find  \(k\)  given  \(\vec a = \left( {\begin{array}{*{20}{c}}
2\\
k
\end{array}} \right)\)  and  \(\vec b = \left( {\begin{array}{*{20}{c}}
5\\
1
\end{array}} \right)\)  are perpendicular.

Find a line parallel to \( – \vec i + 4\vec j\) which cuts the \(x\)-axis at 3 using a Cartesian equation.

Find the coordinates \(\left( {a,b,c} \right)\) of the point where the line with parametric equations \(x = – 2 + 2t,{\rm{ }}y = – 3 – 3t\) and \(z = -4 + 2t\) meets the \(YOZ\) plane.

Find the acute angle, correcting to the nearest degree, between the lines:   \(x + 2y = 4\) and \(2x – y = 3\).

Find the acute angle, correcting to the nearest degree, between the lines:   \(x + 2y = 4\) and \(x = – 2y + 3\).

A line has a vector equation \(\left( {\begin{array}{*{20}{c}}
x\\
y
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
3\\
6
\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}
5\\
{ – 4}
\end{array}} \right)\). Find the shortest distance from \(P\left( {2,3} \right)\) to the line, correcting to three significant figures.