Vector equation

Let’s v = (1, 2, 1), u = (0, -1, 3) and w = (1, 0, 7) . Solve the vector equation c1 v + c2 u + c3 w = 0 for variables c1 c2, c3 and decide weather v, u and w are linear dependent or independent

Result

c₁ : (Correct answer is: )
c₂ = 2 c1
c₃ = -c1

Step-by-step explanation:

$$\begin{gathered} c_1\cdot 1+c_2\cdot 0+c_3\cdot 1=0 \\ {c}_1\cdot 2+c_2\cdot (-1)+c_3\cdot 0=0 \\ {c}_1\cdot 1+c_2\cdot 3+c_3\cdot 7=0 \\ {c}_1+c_3=0 \\ 2c_1-c_2=0 \\ {c}_1+3c_2+7c_3=0 \\ {c}_1=any \\ {c}_2=2c_1 \\ {c}_1=c_3=-c_1 \end{gathered}$$

$c_2=2c_1$

$c_3=-c_1$

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