If x ^ , y ^ and z ^ are three unit vectors in threedimensional space, then the minimum value of ∣ x ^ + y ^ ∣ 2 + ∣ y ^ + z ^ ∣ 2 + ∣ z ^ + x ^ ∣ 2

Question

If x ^ , y ^ ​ and z ^ are three unit vectors in threedimensional space, then the minimum value of ∣ x ^ + y ^ ​ ∣ 2 + ∣ y ^ ​ + z ^ ∣ 2 + ∣ z ^ + x ^ ∣ 2

TestHub · Accepted Answer

( x ^ + y ^ ​ + z ^ ) 2 ≥ 0 ​ ⇒ 3 + 2Σ x ^ ⋅ y ^ ​ ≥ 0 ⇒ 2Σ x ^ ⋅ y ^ ​ ≥ − 3 ​ Now, ∣ x ^ + y ^ ​ ∣ 2 + ∣ y ^ ​ + z ^ ∣ 2 + ∣ z ^ + x ^ ∣ 2 = 6 + 2 ∑ x ^ . y ^ ​ ≥ 6 + ( − 3 ) ⇒ ∣ x ^ + y ^ ​ ∣ 2 + ∣ y ^ ​ + z ^ ∣ 2 + ∣ z ^ + x ^ ∣ 2 ≥ 3

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