Science:Math Exam Resources/Courses/MATH152/April 2017/Question A 06/Solution 1

In order to find a normal vector to the plane containing these three points, we need to find two vectors spanning this plane. This can be achieved by subtracting the points:
${\begin{aligned}(4,1,2)-(3,0,1)&=(1,1,1)\\(4,-2,4)-(4,1,2)&=(0,-3,2)\end{aligned}}$
To find a normal vector to the plane, we need to take the cross product of these vectors.
$\left|{\begin{array}{ccc}{\vec {i}}&{\vec {j}}&{\vec {k}}\\1&1&1\\0&-3&2\end{array}}\right|$
$=5{\vec {i}}-2{\vec {j}}-3{\vec {k}}$.
Answer: $\color {blue}(5,-2,-3)$