LinAlg4.2.23

From Example Problems
Jump to: navigation, search

Find the equation of the plane passing through the points 3{\bar  {i}}-5{\bar  {j}}-{\bar  {k}},-{\bar  {i}}+5{\bar  {j}}+7{\bar  {k}}\, and parallel to 3{\bar  {i}}-{\bar  {j}}+7{\bar  {k}}\,

Let \pi \, be the plane passing through the points a=3{\bar  {i}}-5{\bar  {j}}-{\bar  {k}},b=-{\bar  {i}}+5{\bar  {j}}+7{\bar  {k}}\, and parallel to c=3{\bar  {i}}-{\bar  {j}}+7{\bar  {k}}\, and let r be any point in the plane.

Therefore the vector equation is {\bar  {r}}\cdot ({\bar  {b}}\times {\bar  {c}}+{\bar  {c}}\times {\bar  {a}})=[{\bar  {a}}{\bar  {b}}{\bar  {c}}]\,

Now {\bar  {b}}\times {\bar  {c}}={\begin{vmatrix}{\bar  {i}}&{\bar  {j}}&{\bar  {k}}\\-1&5&7\\3&-1&7\end{vmatrix}}={\bar  {i}}(35+7)-{\bar  {j}}(-7-21)+{\bar  {k}}(1-15)=42{\bar  {i}}+28{\bar  {j}}-14{\bar  {k}}\,


{\bar  {c}}\times {\bar  {a}}={\begin{vmatrix}{\bar  {i}}&{\bar  {j}}&{\bar  {k}}\\3&-1&7\\3&-5&-1\end{vmatrix}}={\bar  {i}}(1+35)-{\bar  {j}}(-3-21)+{\bar  {k}}(-15+3)=36{\bar  {i}}+24{\bar  {j}}-12{\bar  {k}}\,

({\bar  {b}}\times {\bar  {c}}+{\bar  {c}}\times {\bar  {a}})=78{\bar  {i}}+52{\bar  {j}}-26{\bar  {k}}=26(3{\bar  {i}}+2{\bar  {j}}-{\bar  {k}})\,

[abc]={\begin{vmatrix}3&-5&-1\\-1&5&7\\3&-1&7\end{vmatrix}}=3(35+7)+5(-7-21)+(-1)(1-15)=126-140+14=0\,

Therefore vector equation is {\bar  {r}}\cdot (78{\bar  {i}}+52{\bar  {j}}-26{\bar  {k}})\, which is equal to {\bar  {r}}\cdot (3{\bar  {i}}+2{\bar  {j}}-{\bar  {k}})=0\,


Main Page:Linear Algebra