Science:Math Exam Resources/Courses/MATH152/April 2022/Question A06/Solution 1

A point on the line is a vector $(9+4t){\vec {i}}+t{\vec {j}}+t{\vec {k}}$. This vector is orthogonal to the line if and only if

$$\left((9+4t){\vec {i}}+t{\vec {j}}+t{\vec {k}}\right)\cdot \left(4{\vec {i}}+{\vec {j}}+{\vec {k}}\right)=0.$$

Solving the above equation, we find $t=-2$, so the point $\left((9-8),-2,-2\right)\in L$ is the point on the line that is closest to the origin. The distance between this point and the origin is

$${\sqrt {1^{2}+2^{2}+2^{2}}}=3,$$

and this is the shortest distance between the line $L$ and the origin.