$ c - 2a = 6 $,

February 22, 2026

$ c - 2a = 6 $,

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\mathbf{v} \times \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix} = \begin{pmatrix} b(2) - c(0) \\ c(1) - a(2) \\ a(0) - b(1) \end{pmatrix} = \begin{pmatrix} 2b \\ c - 2a \\ -b \end{pmatrix} = \begin{pmatrix} -4 \\ 6 \\ 2 \end{pmatrix}.
This gives the system:
$ 2b = -4 \Rightarrow b = -2 $,
$ -b = 2 \Rightarrow b = -2 $ (consistent).
From $ b = -2 $, substitute into 2: $ c = 6 + 2a $. Let $ a $ be a free variable. Choosing $ a = 0 $, then $ c = 6 $. Thus, $ \mathbf{v} = \begin{pmatrix} 0 \\ -2 \\ 6 \end{pmatrix} $. Final answer: $ \boxed{\begin{pmatrix} 0 \\ -2 \\ 6 \end{pmatrix}} $.**Question 1:
The ratio of the number of apples to oranges in a basket is 3:2. If there are 30 fruits in total, how many apples are in the basket?

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