have discussed on the three dimensional right handed rectangular coordinate system. In this system, when the positive x -axis is rotated counterclockwise into the positive y -axis, a right handed (standard) screw would advance in the direction of the positive z -axis (Fig . (i)). In a right handed coordinate system, the thumb of the right hand points in the direction of the positive z -axis when the fingers are curled in the direction away from the positive x -axis toward the positive y -axis (Fig .
(ii)). Fig . (i), (ii) Definition The vector product of two nonzero vectors , is denoted by and defined as where, θ is the angle between , ≤θ ≤π and ˆ n is a unit vector perpendicular to both , such that form a right handed system (Fig . ).
i.e., the right handed system rotated from moves in the direction of ˆ n . If either , then θ is not defined and in this case, we define Observations . is a vector. .
Let be two nonzero vectors. Then if and only if are parallel (or collinear) to each other, i.e., Fig . In particular, , since in the first situation, θ = and in the second one, θ = π , making the value of sin θ to be . .
If π θ = then . In view of the Observations and , for mutually perpendicular unit vectors k (Fig . ), we have × = = ˆ × = . In terms of vector product, the angle between two vectors may be given as sin θ = .
It is always true that the vector product is not commutative, as Indeed, , where form a right handed system, i.e., θ is traversed from , Fig . (i). While, , where form a right handed system i.e. θ is traversed from Fig .
(ii). Fig . (i), (ii) Thus, if we assume to lie in the plane of the paper, then n n both will be perpendicular to the plane of the paper. But, ˆ n being directed