Find the equation of the plane passing through (a,b,c)and parallel to the plane \(\overrightarrow{r}\).(\(\hat i+\hat j+\hat k\))=2.
Any plane parallel to the plane \(\overrightarrow{r} . \hat i+\hat j+\hat k\) =2, is of the form
\(\overrightarrow{r}\).(\(\hat i+\hat j+\hat k\)) = λ...(1)
The plane passes through the point (a,b,c).
Therefore, the position vector r→ of this point is \(\overrightarrow{r}\)=\(a\hat i+b\hat j+c\hat k\)
Therefore, equation(1) becomes
(\(a\hat i+b\hat j+c\hat k\)).(\(\hat i+\hat j+\hat k\))=λ
⇒a+b+c=λ
Substituting λ=a+b+c in equation(1), we obtain
\(\overrightarrow{r}\).(\(\hat i+\hat j+\hat k\))=a+b+c...(2)
This is the vector equation of the required plane.
Substituting\(\overrightarrow{r}\)=\(x\hat i+y\hat j+z\hat k\) in equation(2), we obtain
(\(x\hat i+y\hat j+z\hat k\)).(\(\hat i+\hat j+\hat k\))=a+b+c
⇒x+y+z=a+b+c.
The correct IUPAC name of \([ \text{Pt}(\text{NH}_3)_2\text{Cl}_2 ]^{2+} \) is: