Reciprocal And Dependent Systems

Question

Three unit vectors a, b and c satisfy a + b + c = 0; using this constraint, evaluate the sum a . b + b . c + c . a precisely.

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The decisive identity is |a + b + c|^2 = |a|^2 + |b|^2 + |c|^2 + 2(a . b + b . c + c . a), obtained by expanding the squared magnitude of the sum. Exploiting a vanishing vector sum to extract pairwise dot products is a classic JEE Advanced manoeuvre. Since a + b + c = 0, its magnitude is zero, so the left side is 0. Each vector is a unit vector, so |a|^2 = |b|^2 = |c|^2 = 1, summing to 3. The expansion becomes 0 = 3 + 2(a . b + b . c + c . a), which rearranges to 2(a . b + b . c + c . a) = -3, hence the required sum equals -3/2. Option 3/2 drops the negative sign during rearrangement. Option 0 would falsely assume mutual perpendicularity, impossible for three vectors summing to zero. Option -1 uses an incorrect magnitude sum of 2 instead of 3. Geometrically the three unit vectors form a closed equilateral triangle with 120-degree angles between each pair, and cos(120) = -1/2 gives three terms of -1/2 each, totalling -3/2. Plausibility check: 3 times (-1/2) equals -3/2, matching the algebraic result exactly.

Reciprocal And Dependent Systems

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