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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