Science:Math Exam Resources/Courses/MATH152/April 2013/Question A 03/Solution 1

$a$ and $d$ are orthogonal whenever their dot product is equal to zero. That is,

$\begin{array}{rl}0=a\cdot d& =\left[\begin{array}{c}-3\\ -3\\ 3\end{array}\right]\cdot \left[\begin{array}{c}-1\\ k\\ 2k+1\end{array}\right]\\ & =(-3)(-1)+(-3)(k)+(3)(2k+1)\\ & =3-3k+6k+3\\ & =6+3k\\ & =3(2+k).\end{array}$

Therefore, $k=-2$ is the value for which $a$ and $d$ are orthogonal.