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Intuit

In the real plane \(\mathbb{R}^2\):

1. Draw \(\vec{v}\) as a point \(P\).
2. Measure the length from the origin to \(P\).

The correspondence with Pythagorean theorem:

1. Recall that \(c^2 = a^2 + b^2\) and so \(c = \sqrt{a^2 + b^2}\).
2. Set \(a = v_1\), \(b = v_2\), and \(c = \|\vec{v}\|\).
3. Get \(\|\vec{v}\| = \sqrt{{v_1}^2 + {v_2}^2}\).