A particle of mass \( 1 \, \text{kg} \) is subjected to a force which depends on the position as \( \vec{F} = -k(x\hat{i}+y\hat{j}) \, \text{kg m s}^{-2} \) with \( k = 1 \, \text{kg s}^{-2} \). At time \( t = 0 \), the particle’s position \( {r} = \left( \dfrac{1}{\sqrt{2}}\hat{i} + \sqrt{2}\hat{i} \right) \, \text{m} \) and its velocity \( {v} = \left( -\sqrt{2}\hat{i} + \sqrt{2}\hat{j} + \dfrac{2}{\pi}\hat{k} \right) \, \text{m s}^{-1} \). Let \( v_x \) and \( v_y \) denote the \( x \) and the \( y \) components of the particle’s velocity, respectively. Ignore gravity. When \( z = 0.5 \, \text{m} \) the value of \( (x v_y - y v_x) \) is _____ \( \text{m}^2 \text{s}^{-1} \).
