What is cross-product and dot-product?

Answer 1

Cross products and dot products are two operations that can be done on a pair of 2-dimensional, 3-dimensional, or n-dimensional vectors. Both can be viewed in terms of mathematics or their physical representations.

The dot product of two three-dimensional vectors A= and B= is a1b1+ a2b2 + a3b3. The definition in high dimensions is completely analogous. Notice that the dot product of two vectors is a scalar, not a vector. The dot product also equals |A|*|B|cosθ, where |A| and |B| are the magnitudes of A and B, respectively and θ is the angle between the vectors. This is the same as saying that the dot product is the magnitude of one vector multiplied times the component of the second vector that is parallel to the first. Notice that this means that the dot product of two vectors is 0 if and only if they are perpendicular.

The cross product is a little more complicated. In three dimensions, A × B = <a2b3 − a3b2, a3b1 − a1b3, a1b2 − a2b1>. Notice that this operation results in another vector. This vector always points in a direction perpendicular to both A and B, and this direction can be determined by the right-hand rule. Physically, the magnitude of this vector equals |A|*|B|sinθ, or the magnitude of the first vector times the component of the other that is perpendicular to the first. So the cross product is 0 when the vectors are parallel.