a=Ai+Bj+Ckb=Di+Ej+Fk{\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}}} Here, i,j,k{\displaystyle \mathbf {i} ,\mathbf {j} ,\mathbf {k} } are unit vectors, and A,B,C,D,E,F{\displaystyle A,B,C,D,E,F} are constants.
a×b=|ijkABCDEF|{\displaystyle \mathbf {a} \times \mathbf {b} ={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \A&B&C\D&E&F\end{vmatrix}}}
a×b=(BF−EC)i−(AF−DC)j+(AE−DB)k{\displaystyle \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} } and b. {\displaystyle \mathbf {b} . }
u=2i−j+3kv=5i+7j−4k{\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}}}
u×v=|ijk2−1357−4|{\displaystyle \mathbf {u} \times \mathbf {v} ={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \2&-1&3\5&7&-4\end{vmatrix}}}
u×v=(4−21)i−(−8−15)j+(14+5)k=−17i+23j+19k{\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}}}
