when will the cross product of two vectors be zero
Cross product If two vectors have the same direction (or have the exact opposite direction from one another, i.e. are not linearly independent) or if either one has zero length, then their cross product is zero. More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths. The cross product is anticommutative (i.e.,   a × b = − b × a  {\displaystyle \mathbf {a} \times \mathbf {b} =-\mathbf {b} \times \mathbf {a} } ) and is distributive over addition (i.e.,   a × ( b + c ) = a × b + a × c  {\displaystyle \mathbf {a} \times (\mathbf {b} +\mathbf {c} )=\mathbf {a} \times \mathbf {b} +\mathbf {a} \times \mathbf {c} } ). The space    R  3   {\displaystyle \mathbb {R} ^{3}}  together with the cross product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket.
Cross River State Cross River is a state in South South Nigeria,[2]:222 bordering Cameroon to the east. Its capital is Calabar, and its name is derived from the Cross River (Oyono), which passes through the state. Boki, Ejagham and Efik are major languages of this state.