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In the triangle ΔPQR, the vectors →a, →b, and →c represent →QR, →RP, and →PQ respectively. If the magnitudes are |→a| = 12 and |→b| = 4√3, and the dot product →b ⋅ →c equals 24, which of the following statements is correct?

1. |→c|² / 2 − |→a|² equals 12
2. |→c|² / 2 − |→a|² equals 30
3. (→a × →b) ⋅ →c equals 48√3
4. The dot product →a ⋅ →b equals −72

Correct answer: The dot product →a ⋅ →b equals −72

Solution

Since a=QR, b=RP, c=PQ close the triangle, a+b+c=0, so c=-(a+b). Then b.c = b.(-a-b) = -a.b - |b|^2 = 24 with |b|^2=(4sqrt3)^2=48, giving a.b = -72. So the correct statement is a.b = -72 (index 3); the stored option (|c|^2/2 - |a|^2 = 12) evaluates to -120 and is wrong.

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