We have,
Given that equation of planes are
→r.(2ˆi+2ˆj−3ˆk)=7......(1)
→r.(2ˆi+5ˆj+3ˆk)=9......(2)
And through the point (2,1,3).
So, we know that,
On comparing that, equation (1) and (2) to, we get,
→r.→n1=d1 and→r.→n2=d2
n1=2ˆi+2ˆj−3ˆkandd1=7
n2=2ˆi+5ˆj+3ˆkandd2=9
So, equation of plane is
→r.(→n1+λ→n2)=d1+λd2
⇒→r.[(2ˆi+2ˆj−3ˆk)+λ(2ˆi+5ˆj+3ˆk)]=7+9λ
⇒→r.[2ˆi+2ˆj−3ˆk+2λˆi+5λˆj+3λˆk]=9λ+7
⇒→r.[(2+2λ)ˆi+(2+5λ)ˆj+(−3+3λ)ˆk]=9λ+7......(3)
Now, to find λ, put →r=xˆi+yˆj+zˆk
So,
(xˆi+yˆj+zˆk).[(2+2λ)ˆi+(2+5λ)ˆj+(−3+3λ)ˆk]=9λ+7
x(2+2λ)ˆi+y(2+5λ)ˆj+z(−3+3λ)ˆk=9λ+7
But the plane passes
through the point (2,1,3).
Now,
2(2+2λ)+1(2+5λ)+3(−3+3λ)=9λ+7
⇒4+4λ+2+5λ−9+9λ=9λ+7
⇒18λ−9λ=7+3
⇒9λ=10
⇒λ=109
Putting value of λ in equation (3) and we get,
→r.[(2+2λ)ˆi+(2+5λ)ˆj+(−3+3λ)ˆk]=9λ+7
⇒→r.[(2+209)ˆi+(2+509)ˆj+(−3+309)ˆk]=10+7
⇒→r.[389ˆi+689ˆj+39ˆk]=17
⇒19→r.[38ˆi+68ˆj+3ˆk]=17
⇒→r.[38ˆi+68ˆj+3ˆk]=17×9
⇒→r.[38ˆi+68ˆj+3ˆk]=153
Hence, this is the
answer.