Geometry -- 1

Let $a = |\cos \alpha, \sin \alpha|$ , $b= |\cos \beta, \sin \beta|$ $\\$ $a + 3b = |\cos \alpha + 3\cos \beta, \sin \alpha + 3 \sin \beta|$ $\\$ $|a + 3b| = \sqrt {\cos^2 \alpha + 9\cos^2 \beta + 6\cos \alpha \cos \beta + \sin^2 \alpha + 9\sin^2 \beta + 6 \sin \alpha \sin \beta}$ $\\$ $|a + 3b| = \sqrt {1- + 6\cos (\alpha - \beta)} = \sqrt {13}$