If A → · B → = A → · C → , does it imply that B → = C → ?

Given, A → · B → = A → · C → A → · B → - A → · C → = 0 A → · ( B → - C → ) = 0 | A → | | ( B → - C → ) | cos θ = 0 Now, either A → = 0 or ( B → - C → ) = 0 or cos θ = 0 If A → ≠ 0 and cos θ ≠ 0 , then we can say ( B → - C → ) = 0 or B → = C → Otherwise, it is not always necessary that B → = C → .

If A → × B → = A → × C → , does it imply that B → = C → ?

Given, we know that A → × B → = A → × C → ⇒ A → × B → - C → = 0 ⇒ A → = 0 o r B → - C → = 0 Always remember that if we want to divide both sides by A, we have to check that we are not dividing by 0 . Also, by right-hand rule, it is understood that the vector product does not obey the commutative property.