LinAlg4.2.19

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Find the angle between the planes {\bar  {r}}\cdot (2{\bar  {i}}-{\bar  {j}}+{\bar  {k}})=6,{\bar  {r}}\cdot ({\bar  {i}}+{\bar  {j}}+2{\bar  {k}})=7\,

Given vector equations of the planes are {\bar  {r}}\cdot (2{\bar  {i}}-{\bar  {j}}+{\bar  {k}})=6,{\bar  {r}}\cdot ({\bar  {i}}+{\bar  {j}}+2{\bar  {k}})=7\,

where first one is m1 and the second one is m2.

Let \theta \, be the angle between the given planes.

Therefore, \cos \theta ={\frac  {m_{1}\cdot m_{2}}{|m_{1}||m_{2}|}}\,

\cos \theta ={\frac  {(2{\bar  {i}}-{\bar  {j}}+{\bar  {k}})\cdot ({\bar  {i}}+{\bar  {j}}+2{\bar  {k}})}{{\sqrt  {4+1+1}}\cdot {\sqrt  {1+1+4}}}}\,

Hence \cos \theta ={\frac  {2-1+2}{{\sqrt  {6}}{\sqrt  {6}}}}={\frac  {3}{6}}={\frac  {1}{2}}=\cos 60\,

Therefore \theta =60^{\circ }\,


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