Question

The two vectors $\left[\begin{array}{ccc}1& 1& 1\end{array}\right]$ and $\left[\begin{array}{ccc}1& a& a^2\end{array}\right]$ where a= $-\frac{1}{2}+j\frac{\sqrt{3}}{2}$ and j = $\sqrt{-1}$ are

• A. orthonormal
• B. orthogonal
• C. parallel
• D. collinear

Answer 1

The correct option is B orthogonal

Let $X_1=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]and{X}_2=\left[\begin{array}{c}1\\ a\\ a^2\end{array}\right]=\left[\begin{array}{c}1\\ w\\ w^2\end{array}\right]$
$\therefore a=\frac{1}{2}+i\frac{\sqrt{3}}{2}=w$ (Cube root of unity)

Now $X_1.X_2=1\times 1+1\times w+1\times w^2$
$=1+w+w^2=0$

Hence they are orthogonal.