The equation of plane passing through given plenes is
\(\vec r((2\hat i + \hat j + \hat k) + \lambda(3\hat i+4\hat j+\hat k))=-4+6\lambda\)
⇒ \(\vec r((2+3\lambda)\hat i+(1+4\lambda)\hat j+(1+\lambda)\hat k)= -4+6\lambda\)
⇒ \((x\hat i+y\hat j+z\hat k).((2+3\lambda)\hat i+(1+4\lambda)\hat j+(1+\lambda)\hat k)=-4+6\lambda\)
(\(\because\hat r = x\hat i+y\hat j+z\hat k\))
⇒ (2 + 3\(\lambda\))x + (1 + 4\(\lambda\))y + (1 + \(\lambda\))z = -4 + 6\(\lambda\)----(1)
It passes through (1, 3, 1)
\(\therefore\) (2 + 3\(\lambda\)).1 + (1 + 4\(\lambda\)).3 + (1 +\(\lambda\)).1 = -4 + 6\(\lambda\)
⇒ 2 + 3\(\lambda\) + 3 + 12\(\lambda\) + 1 + \(\lambda\) = -4 + 6\(\lambda\)
⇒ 10\(\lambda\) + 10 = 0
⇒ \(\lambda\) = -10/10 = -1
\(\therefore\) From(1),
(2 - 3)x + (1 - 4)y + (1 - 1) z = -4 - 6
⇒ -x - 3y = -10
⇒ x + 3y =10
\(\therefore(x\hat i+y\hat j+z\hat k).(\hat i+3\hat j+0\hat k)\) = 10
Hence, vector equation of required plane is
\((x\hat i+y\hat j+z\hat k).(\hat i + 3\hat j+0\hat k)=10\)
