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 elasticity of demand In economics, the cross elasticity of demand or cross-price elasticity of demand measures the responsiveness of the quantity demanded for a good to a change in the price of another good, ceteris paribus. It is measured as the percentage change in quantity demanded for the first good that occurs in response to a percentage change in price of the second good. For example, if, in response to a 10% increase in the price of fuel, the demand for new cars that are fuel inefficient decreased by 20%, the cross elasticity of demand would be:cross body    − 20 %  10 %  = − 2  {\displaystyle {\frac {-20\%}{10\%}}=-2} . A negative cross elasticity denotes two products that are complements, while a positive cross elasticity denotes two substitute products. Assume products A and B are complements, meaning that an increase in the price for A accompanies a decrease in the quantity demanded for B. Therefore, if the price of product B decreases, the demand curve for product A shifts to the right reflecting an increase in A's demand, resulting in a negative value for the cross elasticity of demand. The exact opposite reasoning holds for substitutes.