Let

--------(1) (∵ α is angle between a⃗ and b⃗ )

Then

(∵θ is angle between e⃗ and c⃗ ⇒ θ is angle between a⃗ Xb⃗ and c⃗ )

(∵ using (1))

Hence, if and only if

if and only if and

if and only if and

⇒ a⃗ and b⃗ are perpendicular.

Also e⃗ is perpendicular to both a⃗ and b⃗ .

θ=0⇒ c⃗ is perpendicular to both a⃗ and b⃗

∴ a⃗, b⃗, c⃗ are mutually perpendicular.

∴ a⃗∙ b⃗=b⃗∙ c⃗=c⃗∙ a⃗=0