Question #197379

let U=2i+2j+k/3,V=i-j/√2 & W=-√2(i+j-4k)/6 .compute the scalar products U.V,U.W &V.W . Check whether U,V & W are orthonormal


1
Expert's answer
2021-05-24T17:16:53-0400

U=2i^+2j^+13k^U=2\hat i+2\hat j+\dfrac{1}{3}\hat k

V=i^12j^V=\hat i-\dfrac{1}{\sqrt{2}}\hat j

W=2(i^+j^4k^)/6W=-√2(\hat i+\hat j-4\hat k)/6

U.V=((2×1)+(2×12)+(13×0)=22=0.58U.V=\bigg((2\times1)+(2\times\dfrac{-1}{\sqrt2})+(\dfrac{1}{3}\times0\bigg)=2-\sqrt 2=0.58

U.W=(2×26)+(2×26)+(13×46)=1.37U.W=(2\times\dfrac{-\sqrt 2}{6})+(2\times-\dfrac{\sqrt 2}{6})+(\dfrac{1}{3}\times\dfrac{-4}{6})=-1.37

V.W=(1×26)+(12×26)+(0×426)=0.06V.W=(1\times\dfrac{-\sqrt2}{6})+(-\dfrac{1}{\sqrt2}\times\dfrac{-\sqrt2}{6})+(0\times\dfrac{4\sqrt2}{6})=0.06


Scalar triple product of three vectors, U,V,W=(U×V).WU,V,W=(U\times V).W

=((2×12)k^(2×1)k^+(13×1)j^(13×12)i^).W=\bigg((2\times\dfrac{-1}{\sqrt2})\hat k-(2\times1)\hat k+(\dfrac{1}{3}\times1)\hat j-(\dfrac{1}{3}\times\dfrac{-1}{\sqrt 2})\hat i\bigg).W

=(132i^+13j^+(22)k^).(26(i^+j^4k^)=\bigg(\dfrac{1}{3\sqrt2}\hat i+\dfrac{1}{3}\hat j+(-2-\sqrt2)\hat k\bigg).\bigg(\dfrac{-√2}{6}(\hat i+\hat j-4\hat k\bigg)

=1921882686=-\dfrac{1}{9}-\dfrac{\sqrt2}{18}-\dfrac{8\sqrt2}{6}-\dfrac{8}{6}

= 0\cancel=\space0

Therefore U, V and W are not orthogonal


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