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We need to find the coordinates for three different vectors(\(\mathbf{u}\), \(\mathbf{v}\), \(\mathbf{w}\)) that share the same direction angle, which is \(\frac{\pi}{6}\). Furthermore, we will choose three arbitrary magnitudes to make sure they are distinct.

We can write the general form of the vector as \(\mathbf{u} = u_x\hat{\mathbf{i}} + u_y\hat{\mathbf{j}}\), where \(u_x\) is the magnitude of the \(\mathbf{x}\)-component of the vector, \(u_y\) is the magnitude of the \(\mathbf{y}\)-component, and \(\hat{\mathbf{i}}\) and \(\hat{\mathbf{j}}\) are the unit vectors in the x and y directions, respectively. To find \(u_x\) and \(u_y\), we can use the sine and cosine functions: \[u_x = |\mathbf{u}|\cos{\frac{\pi}{6}}\] \[u_y = |\mathbf{u}|\sin{\frac{\pi}{6}}\] where \(|\mathbf{u}|\) is the magnitude of the vector.

To find three different vectors with the same direction angle but distinct magnitudes, we can choose any three arbitrary non-zero magnitudes, say \(|\mathbf{u}|=2\), \(|\mathbf{v}|=4\), \(|\mathbf{w}|=6\). For \(\mathbf{u}\): \[u_x = 2\cos{\frac{\pi}{6}} = 2\cdot\frac{\sqrt{3}}{2} = \sqrt{3}\] \[u_y = 2\sin{\frac{\pi}{6}} = 2\cdot\frac{1}{2} = 1\] So, \[\mathbf{u} = (\sqrt{3},1)\] For \(\mathbf{v}\): \[v_x = 4\cos{\frac{\pi}{6}} = 4\cdot\frac{\sqrt{3}}{2} = 2\sqrt{3}\] \[v_y = 4\sin{\frac{\pi}{6}} = 4\cdot\frac{1}{2} = 2\] So, \[\mathbf{v} = (2\sqrt{3},2)\] For \(\mathbf{w}\): \[w_x = 6\cos{\frac{\pi}{6}} = 6\cdot\frac{\sqrt{3}}{2} = 3\sqrt{3}\] \[w_y = 6\sin{\frac{\pi}{6}} = 6\cdot\frac{1}{2} = 3\] So, \[\mathbf{w} = (3\sqrt{3},3)\] Now we have three different vectors each with a direction angle of \(\frac{\pi}{6}\) and different magnitudes. They are: \[\mathbf{u} = (\sqrt{3},1), \quad \mathbf{v} = (2\sqrt{3}, 2), \quad \mathbf{w} = (3\sqrt{3}, 3)\]