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

$v_3=v_1\times v_2=[2,1,-1]\times [-1,-2,3]=[1,-5,-3]$

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