If $a=\begin{bmatrix} 2 \\ -2 \\ 1 \end{bmatrix},b=\begin{bmatrix} 2 \\ -2 \\ 5 \end{bmatrix},c=\begin{bmatrix} -1 \\ 1 \\ 2 \end{bmatrix}$, find the values of $\alpha$ and $\beta$ such that $a+\alpha b+\beta c=0$
I'm in engineering school, this is a question from my tutorial. I've read though my lecture notes and have studied, but still absolutely clueless on how to go about it, and what it's asking for. Would appreciate help, sitting for a test tommorow but I'm stuck with this.
Given $$a=\begin{bmatrix} 2 \\ -2 \\ 1 \end{bmatrix},b=\begin{bmatrix} 2 \\ -2 \\ 5 \end{bmatrix},c=\begin{bmatrix} -1 \\ 1 \\ 2 \end{bmatrix}$$ $$a+\alpha b+\beta c=0$$ $$\begin{bmatrix} 2 \\ -2 \\ 1 \end{bmatrix}+\begin{bmatrix} 2\alpha \\ -2\alpha \\ 5\alpha \end{bmatrix}+\begin{bmatrix} -1\beta \\ 1\beta \\ 2\beta \end{bmatrix}=\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$$ $$\begin{bmatrix} 2+2\alpha-\beta \\ -2-2\alpha+\beta \\ 1+5\alpha+2\beta \end{bmatrix}=\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$$ Now solve the equations to get the value of $\alpha$ and $\beta$.
Can you take it from here?