Question

If a is any vector then show that (a*i)² + (a*j)² + (a*k)² = 2a²

Answer 1

Let vector a = a₁i + a₂j + a₃k

vector{(a*i)² + (a*j)² + (a*k)²}

= vector{|a|²|i|² - (a.i)² + |a|²|j|² -(a.j)² + |a|²|j|² - (a.k)²}

= 3 vector{|a|² - [(a.i)² + (a.j)² + (a.k)²] }

= 3 vector a^-2 - (a₁² a₂² a₃² )

= 3 vector{a^-2 - a^-2 }

= 2 vector a^-2