Question

Find the value of $(a\times i{)}^2+(a\times j{)}^2+(a\times K{)}^2$

Answer 1

Let $a=a_{1}i+a_{2}j+a_{3}k$

Then,

$a\times i=-a_{2}k+a_{3}j$

$a\times j=a_{1}k-a_{3}i$

$a\times k=-a_j+a_{2}i$

Therefore,

$(a\times i{)}^2+(a\times j{)}^2+(a\times k{)}^2$

$=2({a_1}^2+{a_2}^2+{a_3}^2)=2a^2$