Calculating the Cross Product
Consider two general three-dimensional vectors defined in Cartesian coordinates. a = A i + B j + C k b = D i + E j + F k {\displaystyle {\begin{aligned}\mathbf {a} &=A\mathbf {i} +B\mathbf {j} +C\mathbf {k} \\\mathbf {b} &=D\mathbf {i} +E\mathbf {j} +F\mathbf {k} \end{aligned}}} {\begin{aligned}{\mathbf {a}}&=A{\mathbf {i}}+B{\mathbf {j}}+C{\mathbf {k}}\\{\mathbf {b}}&=D{\mathbf {i}}+E{\mathbf {j}}+F{\mathbf {k}}\end{aligned}} Here, i , j , k {\displaystyle \mathbf {i} ,\mathbf {j} ,\mathbf {k} } {\mathbf {i}},{\mathbf {j}},{\mathbf {k}} are unit vectors, and A , B , C , D , E , F {\displaystyle A,B,C,D,E,F} A,B,C,D,E,F are constants.
Set up the matrix. One of the easiest ways to compute a cross product is to set up the unit vectors with the two vectors in a matrix. a × b = | i j k A B C D E F | {\displaystyle \mathbf {a} \times \mathbf {b} ={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\A&B&C\\D&E&F\end{vmatrix}}} {\mathbf {a}}\times {\mathbf {b}}={\begin{vmatrix}{\mathbf {i}}&{\mathbf {j}}&{\mathbf {k}}\\A&B&C\\D&E&F\end{vmatrix}}
Calculate the determinant of the matrix. Below, we use cofactor expansion (expansion by minors). a × b = ( B F − E C ) i − ( A F − D C ) j + ( A E − D B ) k {\displaystyle \mathbf {a} \times \mathbf {b} =(BF-EC)\mathbf {i} -(AF-DC)\mathbf {j} +(AE-DB)\mathbf {k} } {\mathbf {a}}\times {\mathbf {b}}=(BF-EC){\mathbf {i}}-(AF-DC){\mathbf {j}}+(AE-DB){\mathbf {k}} This vector is orthogonal to both a {\displaystyle \mathbf {a} } {\mathbf {a}} and b . {\displaystyle \mathbf {b} .} {\mathbf {b}}.
Example
Consider the two vectors below. u = 2 i − j + 3 k v = 5 i + 7 j − 4 k {\displaystyle {\begin{aligned}\mathbf {u} &=2\mathbf {i} -\mathbf {j} +3\mathbf {k} \\\mathbf {v} &=5\mathbf {i} +7\mathbf {j} -4\mathbf {k} \end{aligned}}} {\begin{aligned}{\mathbf {u}}&=2{\mathbf {i}}-{\mathbf {j}}+3{\mathbf {k}}\\{\mathbf {v}}&=5{\mathbf {i}}+7{\mathbf {j}}-4{\mathbf {k}}\end{aligned}}
Set up the matrix. u × v = | i j k 2 − 1 3 5 7 − 4 | {\displaystyle \mathbf {u} \times \mathbf {v} ={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\2&-1&3\\5&7&-4\end{vmatrix}}} {\mathbf {u}}\times {\mathbf {v}}={\begin{vmatrix}{\mathbf {i}}&{\mathbf {j}}&{\mathbf {k}}\\2&-1&3\\5&7&-4\end{vmatrix}}
Calculate the determinant of the matrix. u × v = ( 4 − 21 ) i − ( − 8 − 15 ) j + ( 14 + 5 ) k = − 17 i + 23 j + 19 k {\displaystyle {\begin{aligned}\mathbf {u} \times \mathbf {v} &=(4-21)\mathbf {i} -(-8-15)\mathbf {j} +(14+5)\mathbf {k} \\&=-17\mathbf {i} +23\mathbf {j} +19\mathbf {k} \end{aligned}}} {\begin{aligned}{\mathbf {u}}\times {\mathbf {v}}&=(4-21){\mathbf {i}}-(-8-15){\mathbf {j}}+(14+5){\mathbf {k}}\\&=-17{\mathbf {i}}+23{\mathbf {j}}+19{\mathbf {k}}\end{aligned}}
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