Cross product

In mathematics, the cross product or vector product (occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space ( R 3 ) {displaystyle left(mathbb {R} ^{3} ight)} and is denoted by the symbol × {displaystyle imes } . Given two linearly independent vectors a {displaystyle mathbf {a} } and b {displaystyle mathbf {b} } , the cross product, a × b {displaystyle mathbf {a} imes mathbf {b} } (read 'a cross b'), is a vector that is perpendicular to both a {displaystyle mathbf {a} } and b {displaystyle mathbf {b} } and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product).Hence, the left hand side equals C 1 = [ 0 0 0 0 0 1 0 − 1 0 ] , C 2 = [ 0 0 − 1 0 0 0 1 0 0 ] , C 3 = [ 0 1 0 − 1 0 0 0 0 0 ] {displaystyle mathbf {C} _{1}={egin{bmatrix}0&0&0\0&0&1\0&-1&0end{bmatrix}},quad mathbf {C} _{2}={egin{bmatrix}0&0&-1\0&0&0\1&0&0end{bmatrix}},quad mathbf {C} _{3}={egin{bmatrix}0&1&0\-1&0&0\0&0&0end{bmatrix}}} First, that which concerns addition and the scalar and vector products of vectors. Second, that which concerns the differential and integral calculus in its relations to scalar and vector functions. Third, that which contains the theory of the linear vector function. In mathematics, the cross product or vector product (occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space ( R 3 ) {displaystyle left(mathbb {R} ^{3} ight)} and is denoted by the symbol × {displaystyle imes } . Given two linearly independent vectors a {displaystyle mathbf {a} } and b {displaystyle mathbf {b} } , the cross product, a × b {displaystyle mathbf {a} imes mathbf {b} } (read 'a cross b'), is a vector that is perpendicular to both a {displaystyle mathbf {a} } and b {displaystyle mathbf {b} } and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection 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} imes mathbf {b} =-mathbf {b} imes mathbf {a} } ) and is distributive over addition (i.e., a × ( b + c ) = a × b + a × c {displaystyle mathbf {a} imes (mathbf {b} +mathbf {c} )=mathbf {a} imes mathbf {b} +mathbf {a} imes 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. Like the dot product, it depends on the metric of Euclidean space, but unlike the dot product, it also depends on a choice of orientation or 'handedness'. The product can be generalized in various ways; it can be made independent of orientation by changing the result to pseudovector, or in arbitrary dimensions the exterior product of vectors can be used with a bivector or two-form result. Also, using the orientation and metric structure just as for the traditional 3-dimensional cross product, one can in n {displaystyle n} dimensions take the product of n − 1 {displaystyle n-1} vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions. (See § Generalizations, below, for other dimensions.) The cross product of two vectors a and b is defined only in three-dimensional space and is denoted by a × b. In physics, sometimes the notation a ∧ b is used, though this is avoided in mathematics to avoid confusion with the exterior product. The cross product a × b is defined as a vector c that is perpendicular (orthogonal) to both a and b, with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span. The cross product is defined by the formula where θ is the angle between a and b in the plane containing them (hence, it is between 0° and 180°), ‖a‖ and ‖b‖ are the magnitudes of vectors a and b, and n is a unit vector perpendicular to the plane containing a and b in the direction given by the right-hand rule (illustrated). If the vectors a and b are parallel (i.e., the angle θ between them is either 0° or 180°), by the above formula, the cross product of a and b is the zero vector 0. By convention, the direction of the vector n is given by the right-hand rule, where one simply points the forefinger of the right hand in the direction of a and the middle finger in the direction of b. Then, the vector n is coming out of the thumb (see the adjacent picture). Using this rule implies that the cross product is anti-commutative, i.e., b × a = −(a × b). By pointing the forefinger toward b first, and then pointing the middle finger toward a, the thumb will be forced in the opposite direction, reversing the sign of the product vector. Using the cross product requires the handedness of the coordinate system to be taken into account (as explicit in the definition above). If a left-handed coordinate system is used, the direction of the vector n is given by the left-hand rule and points in the opposite direction.