Lagrange Identity

The Lagrange identity states that |a . b|^2 + |a x b|^2 = |a|^2 |b|^2, since the squared cosine and squared sine of the included angle add to one. This elegant relation often shortcuts JEE Advanced problems that appear to demand separate dot and cross computations. By definition |a . b|^2 = |a|^2|b|^2 cos^2(theta) and |a x b|^2 = |a|^2|b|^2 sin^2(theta); adding them and factoring gives |a|^2|b|^2(cos^2 theta + sin^2 theta) = |a|^2|b|^2. Thus the angle becomes irrelevant, and the sum equals (4^2)(3^2) = 16 times 9 = 144. Option 108 is obtained by computing only the cross-product square, 144 times sin... actually 144 times (1/4) plus an error. Option 36 comes from |a|^2 |b|^2 without one factor, using 4 times 9. Option 72 halves the correct product mistakenly. The fact that the 30-degree angle does not appear in the final answer is the central insight the problem rewards. Plausibility check: computing directly, |a . b|^2 = (12 cos 30)^2 = (6 sqrt 3)^2 = 108 and |a x b|^2 = (12 sin 30)^2 = 6^2 = 36, and 108 + 36 = 144, confirming the identity.