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# Find the real values of k for which the given vectors are orthogonal. If there are no such values, show why.$\left[\begin{array}{l} {k} \\ {0} \\ {k^{2}} \end{array}\right],\left[\begin{array}{l} {1} \\ {2} \\ {3} \end{array}\right]$

Solution

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Two vectors are orthogonal if their scalar product is zero:

$\begin{bmatrix} k & 0 & k^2 \end{bmatrix} \cdot\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} =(k)(1)+(0)(2)+(k^2)(3)=k+3k^2=k(1+3k)=0$

Thus we then obtain $k=0$ or $1+3k=0$ which is equivalent with $3k=-1$ or $k=-\dfrac{1}{3}$.

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