If |hata-hatb|=sqrt2∣∣ˆa−ˆb∣∣=√2 then calculate the value of |hata+sqrt3hatb|∣∣ˆa+√3ˆb∣∣?

1 Answer
Cesareo R.
May 29, 2017

22

Explanation:

norm( hat a-hat b)^2=norm(hat a)^2-2 << hat a, hat b >> + norm(hat b)^2 =1-2 << hat a, hat b >>+1 = (sqrt2)^2=2∥∥ˆa−ˆb∥∥2=∥ˆa∥2−2⟨ˆa,ˆb⟩+∥∥ˆb∥∥2=1−2⟨ˆa,ˆb⟩+1=(√2)2=2

so << hat a, hat b >> = 0⟨ˆa,ˆb⟩=0 the unit vectors are orthogonal.

Now

norm( hat a+sqrt3 hat b)^2 = norm( hat a)^2+2 sqrt3 << hat a, hat b >> + 3 norm(hat b)^2 = 1+2sqrt3 xx 0+3 = 4∥∥ˆa+√3ˆb∥∥2=∥ˆa∥2+2√3⟨ˆa,ˆb⟩+3∥∥ˆb∥∥2=1+2√3×0+3=4

so

norm( hat a+sqrt3 hat b)=sqrt(4)=2∥∥ˆa+√3ˆb∥∥=√4=2

NOTE: << cdot, cdot >>⟨⋅,⋅⟩ indicates the scalar product of two vectors.

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