Answer to Question #197379 in Linear Algebra for Subham

Question

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

Answer 1

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

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

$W=-√2(\hat i+\hat j-4\hat k)/6$

$U.V=\bigg((2\times1)+(2\times\dfrac{-1}{\sqrt2})+(\dfrac{1}{3}\times0\bigg)=2-\sqrt 2=0.58$

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

$V.W=(1\times\dfrac{-\sqrt2}{6})+(-\dfrac{1}{\sqrt2}\times\dfrac{-\sqrt2}{6})+(0\times\dfrac{4\sqrt2}{6})=0.06$

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

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

$=\bigg(\dfrac{1}{3\sqrt2}\hat i+\dfrac{1}{3}\hat j+(-2-\sqrt2)\hat k\bigg).\bigg(\dfrac{-√2}{6}(\hat i+\hat j-4\hat k\bigg)$

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

$\cancel=\space0$

Therefore U, V and W are not orthogonal