If (\vec{a} = x\hat{i} + 2\hat{j} - z\hat{k}) and (\vec{b} = 3\hat{i} - y\hat{j} + \hat{k}) are two equal vectors, then write the value of x+y+z.

Equate the vectors (\vec{a} = x\hat{i} + 2\hat{j} - z\hat{k}) and (\vec{b} = 3\hat{i} - y\hat{j} + \hat{k}) by writing them in the component form that is (\vec{a} = \langle x, 2, -z \rangle) and (\vec{b} = \langle 3, -y, 1 \rangle) Then equate the vectors as follows: (\langle x, 2, -z \rangle = \langle 3, -y, 1 \rangle) implies x=3, y=-2, z=-1 Therefore, x+y+z = 3 + (-2) + (-1) = 0 Hence, x+y+z=0.