Find the acute angle between the planes with equations $x + y + z = 3$ and $2x - z = 2$.

n$_1 = \left( {\begin{array}{*{20}{c}} 1 \\ 1 \\ 1 \end{array}} \right)$ and n$_2 = \left( {\begin{array}{*{20}{c}} 2 \\ 0 \\ { - 1} \end{array}} \right)$     (A1)(A1)

EITHER

$\theta  = \arccos \left( {\frac{{{n_1} \bullet {n_2}}}{{\left| {{n_1}} \right|\left| {{n_2}} \right|}}} \right)\left( {\cos \theta  = \frac{{{n_1} \bullet {n_2}}}{{\left| {{n_1}} \right|\left| {{n_2}} \right|}}} \right)$    (M1)

$= \arccos \left( {\frac{{2 + 0 - 1}}{{\sqrt 3 \sqrt 5 }}} \right)\left( {\cos \theta  = \frac{{2 + 0 - 1}}{{\sqrt 3 \sqrt 5 }}} \right)$    (A1)

$= \arccos \left( {\frac{1}{{\sqrt {15} }}} \right)\left( {\cos \theta  = \frac{1}{{\sqrt {15} }}} \right)$

OR

$\theta  = \arcsin \left( {\frac{{\left| {{n_1} \times {n_2}} \right|}}{{\left| {{n_1}} \right|\left| {{n_2}} \right|}}} \right)\left( {\sin \theta  = \frac{{\left| {{n_1} \times {n_2}} \right|}}{{\left| {{n_1}} \right|\left| {{n_2}} \right|}}} \right)$    (M1)

$= \arcsin \left( {\frac{{\sqrt {14} }}{{\sqrt 3 \sqrt 5 }}} \right)\left( {\sin \theta  = \frac{{\sqrt {14} }}{{\sqrt 3 \sqrt 5 }}} \right)$    (A1)

$= \arcsin \left( {\frac{{\sqrt {14} }}{{\sqrt {15} }}} \right)\left( {\sin \theta  = \frac{{\sqrt {14} }}{{\sqrt {15} }}} \right)$

THEN

$= 75.0^\circ {\text{ (or 1.31)}}$    A1

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