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← JEE Main 2014 · 12 Apr · Shift 1
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JEE Main 2014 · 12 Apr · Shift 1, Mathematics question 85

If x^\hat x, y^\hat y and z^\hat z are three unit vectors in three-dimensional space, then the minimum value of
∣x^+y^∣2+∣y^+z^∣2+∣z^+x^∣2|\hat x+\hat y|^2+|\hat y+\hat z|^2+|\hat z+\hat x|^2
is :

Choose one option
Answer and worked solution

Official answer: B

The sum is 6+2(x^⋅y^+y^⋅z^+z^⋅x^)6+2(\hat x\cdot\hat y+\hat y\cdot\hat z+\hat z\cdot\hat x). Since ∣x^+y^+z^∣2=3+2(… )≥0|\hat x+\hat y+\hat z|^2=3+2(\dots)\ge0, the dot-product sum is at least −32-\frac32. So the minimum is 6−3=36-3=3.

Topics:Dot product
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