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,

${\begin{aligned}0=a\cdot d&={\begin{bmatrix}-3\\-3\\3\end{bmatrix}}\cdot {\begin{bmatrix}-1\\k\\2k+1\end{bmatrix}}\\&=(-3)(-1)+(-3)(k)+(3)(2k+1)\\&=3-3k+6k+3\\&=6+3k\\&=3(2+k).\end{aligned}}$

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