Science:Math Exam Resources/Courses/MATH200/April 2012/Question 01 (a)/Solution 2

This quicker alternative solution relies on the cross product: Given two vectors $v_{1},v_{2}$, their cross product $v_{3}=v_{1}\times v_{2}$ is perpendicular to both, $v_{1}$ and $v_{2}$.

We are asked to find a vector that is parallel to a plane, and perpendicular to a line. By the definition of the normal vector of a plane, all vector parallel to a plane are perpendicular to the normal vector. Further, vectors are perpendicular to a line if they are perpendicular to the direction vector of that line. Hence, choose $v_{1}$ as the normal vector of the plane, $v_{1}=[2,1,-1]$, and choose $v_{2}$ as the direction vector of the line, $v_{2}=[-1,-2,3]$. Then

$\displaystyle v_{3}=v_{1}\times v_{2}=[2,1,-1]\times [-1,-2,3]=[1,-5,-3]$

is parallel to the plane, and perpendicular to the line.