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Find the equation of the plane passing through the point (3,4,5)\, and parallel to the plane {\bar  {r}}\cdot (2{\bar  {i}}+3{\bar  {j}}-{\bar  {k}})=6\,

Given plane is {\bar  {r}}\cdot (2{\bar  {i}}+3{\bar  {j}}-{\bar  {k}})=6\,

Let \pi \, be the required plane.Given this is parallel to the above plane.

But \pi \, passes through (3,4,5)\, and is perpendicular to 2{\bar  {i}}+3{\bar  {j}}-{\bar  {k}}\,

Therefore vector equation is ({\bar  {r}}-{\bar  {a}})\cdot {\bar  {n}}=0\,

[{\bar  {r}}-(3{\bar  {i}}+4{\bar  {j}}+5{\bar  {k}})]\cdot (2{\bar  {i}}+3{\bar  {j}}-{\bar  {k}})=0\,

Simplifying we get

{\bar  {r}}\cdot (2{\bar  {i}}+3{\bar  {j}}-{\bar  {k}})=6+12-5=13\,

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