The Cross Product When discussing the dot product, we showed how two vectors can be combined to get a number. Here, we shall see another way of combining vectors, this time resulting in a vector. This operation is called the cross product. The cross product of two vectors vand wis a third vector which is perpendicular to bothvand w. Unlike the dot product (which is defined for any dimension of the vector), the cross product is only defined for three-dimensional vectors. The cross product is more easily remembered in terms of a linear algebra result, as the determinant of a 3 μ3 matrix. Cross Product Let 123vvvvijkand 123wwwwijk. The cross product of vand w, denoted by vμw, is a vectorgiven by 123233231131221123()()(vvvv wv wv wv wv wv wwwwijkvwi)jkExample 1:Compute vμwif 430vijkand 225 wijk. Verify that vμwis perpendicular to both vand w. Solution:vμw= [(–3)(5) – (0)(2)]i+ [(0)(–2) – (4)(5)]j+ [(4)(2) – (–3)(–2)]k= –15i– 20j+ 2k. Notice that and . (4)( 15)( 3)( 20)(0)(2)606000 vvw2)( 15)(2)( 20)(5)(2)3040100(wvwAnd so, we see that both vand ware perpendicular to vμw. Similar to the case of the dot product, there is a nice geometrical interpretation of the cross product, this time in terms of areas of parallelograms. To that end, we make the following claim. 1
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