Question

For any vector a show that $|\overline{a}\times \overline{i}{|}^2+|\overline{a}\times \overline{j}{|}^2+|\overline{a}\times \overline{k}{|}^2=2|\overline{a}{|}^2$

Answer 1

$|\overline{a}\times i{|}^2+|\overline{a}\times j{|}^2+\overline{a}\times k{|}^2$

$=|\overline{a}{|}^2{\sin }^2(a,i)+|\overline{a}{|}^2{\sin }^2(a,j)+|\overline{a}{|}^2{\sin }^2(a,k)$

$=|\overline{a}{|}^2(1-{\cos }^2\alpha +1-{\cos }^2\beta +1-{\cos }^2\gamma )$

$=|\overline{a}{|}^2(3-1)=2|\overline{a}{|}^2$