Let the coordinates of a point P with respect to a system of non-coplanar vectors $\vec a, \vec b$ and $\vec c$ be (3, 2, 1). Then, the coordinates of P with respect to the system of vectors $\vec a+\vec b+\vec c, \vec a-\vec b+\vec c$ and $\vec a+\vec b-\vec c$ are

(3/2, 1/2, 1)

Let the coordinates of P with respect to the new system be (x, y, z). Then,

$3\vec a+2\vec b+\vec c=x(\vec a+\vec b+\vec c) + y (\vec a−\vec b+\vec c) +z (\vec a + \vec b −\vec c)$

$⇒3\vec a+2\vec b+\vec c=(x+y+z) \vec a+ (x−y + z) \vec b + (x + y-z)\vec c$

$⇒x + y + z=3, x-y+z=2$ and $x + y-z=1$  [∵ $\vec a, \vec b, \vec c$ are non-coplanar vectors]

$⇒x=\frac{3}{2},y=\frac{1}{2},z=1$.

Hence, the coordinates of P with respect to the new system are (3/2, 1/2, 1)