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Suppose we have 3 vectors $(a, b, c)$ in the x-y plane and a fourth $(d)$ in the y-z plane. If 4 vectors in 3D space are always linearly dependent, how do we express the fourth in terms of the other 3?

I understand that in writing $xa + yb + zc + md = 0$, we can equate m to 0 and say that the equation is true, but that feels like a trivial way to resolve this.

Is there any better way to show linear dependency directly through $i$, $j$ and $k$ components?

**Please use only vectors as I am not familiar with the connection to matrices.

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While 4 vectors in a 3-dimensional space are linearly dependent, it does not mean that the fourth vector can be expressed through the first three; the dependence means only that some of the 4 vectors can be expressed as a linear combination of the others.

For example, take $a = (1,0,0)$, $b = (0,1,0)$, $c = (1,1,0)$, $d = (0,0,1)$. The only linear relation between these vectors is $a + b - c = 0$ (up to scaling), and it is impossible to express the vector $d$ in terms of $a,b,c$ because every linear combination of $a,b,c$ has zero in the last coordinate.

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You don't (can't) express the fourth in terms of the first three. You find a linear dependence among the first three, which expresses one of them in terms of the other two.

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