The distance between the line r=2 i-2 j+3 k+( i- j+4 k) and the plane r( i+5 j+ k)=5 is

Question

The distance between the line $\vec r=2\hat i-2\hat j+3\hat k+\lambda(\hat i-\hat j+4\hat k)$ and the plane $\vec r\cdot(\hat i+5\hat j+\hat k)=5$ is

Accepted Answer

The plane's normal $(1,5,1)$ is perpendicular to the line's direction $(1,-1,4)$ (dot product $1-5+4=0$), confirming the line is parallel to the plane. Distance from the point $(2,-2,3)$ on the line:
$$\text{Distance}=\frac{|2-10+3-5|}{\sqrt{1+25+1}}=\frac{10}{\sqrt{27}}=\frac{10}{3\sqrt3}.$$

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