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Let a, b and c be three unit vectors such that a + b + c = 0. If λ = a·b + b·c + c·a and d = a×b + b×c + c×a, then the ordered pair (λ, d) is equal to: (1) (-3/2, 3c×b) (2) (-3/2, 3a×b) (3) (3/2, 3b×c) (4) (3/2, 3a×c)

1. (1) (-3/2, 3c×b)
2. (2) (-3/2, 3a×b)
3. (3) (3/2, 3b×c)
4. (4) (3/2, 3a×c)

Correct answer: (2) (-3/2, 3a×b)

Solution

The value of λ is calculated as the sum of the dot products of the unit vectors, which results in -3/2 due to the relationship a + b + c = 0. The vector d, being the sum of the cross products, simplifies to 3a×b, confirming that option (2) is correct.

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