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.
Unit vector The unit vectors appropriate to spherical symmetry are:    r ^   {\displaystyle \mathbf {\hat {r}} } , the direction in which the radial distance from the origin increases;    φ ^   {\displaystyle {\boldsymbol {\hat {\varphi }}}} , the direction in which the angle in the x-y plane counterclockwise from the positive x-axis is increasing; and    θ ^   {\displaystyle {\boldsymbol {\hat {\theta }}}} , the direction in which the angle from the positive z axis is increasing. To minimize redundancy of representations, the polar angle   θ  {\displaystyle \theta }  is usually taken to lie between zero and 180 degrees. It is especially important to note the context of any ordered triplet written in spherical coordinates, as the roles of    φ ^   {\displaystyle {\boldsymbol {\hat {\varphi }}}}  and    θ ^   {\displaystyle {\boldsymbol {\hat {\theta }}}}  are often reversed. Here, the American "physics" convention[1] is used. This leaves the azimuthal angle   φ  {\displaystyle \varphi }  defined the same as in cylindrical coordinates. The Cartesian relations are: