Hipi Zhdripi i Matematikës/1158

i cili mund të shkruhet edhe në formë të përcaktorit simbolik të rendit të tretë në këtë mënyrë: ${\vec {a}}\times {\vec {b}}={\begin{vmatrix}{\vec {i}}&{\vec {j}}&{\vec {k}}\\x_{1}&y_{1}&z_{1}\\x_{2}&y_{2}&z_{2}\\\end{vmatrix}}$ (...25a) ndërkaq sinusi i këndit ndërmjet atyre vektorëve me formulën: $sin({\vec {a}},{\vec {b}})={\sqrt {\frac {{\begin{vmatrix}y_{1}&z_{1}\\y_{2}&z_{2}\end{vmatrix}}^{2}+{\begin{vmatrix}x_{1}&z_{1}\\x_{2}&z_{2}\end{vmatrix}}^{2}+{\begin{vmatrix}x_{1}&y_{1}\\x_{2}&y_{2}\end{vmatrix}}^{2}}{(x_{1}^{2}+y_{1}^{2}+z_{1}^{2})(x_{2}^{2}+y_{2}^{2}+z_{2}^{2})}}}$ (...26a)         Ligjet e prodhimit vektorial të vektorëve vërtetohen fare lehtë kur vektorët janë shprehur me koordinata. Për shembull ligji distributiv vërtetohet kështu: $\!(\!{\vec {a}}\!+\!{\vec {b}}\!)\!\times \!{\vec {c}}\!$ $={\begin{vmatrix}{\vec {i}}&{\vec {j}}&{\vec {k}}\\x_{1}+x_{2}&y_{1}+y_{2}&z_{1}+z_{2}\\x_{3}&y_{3}&z_{3}\end{vmatrix}}$ $={\vec {i}}{\begin{vmatrix}y_{1}\!+\!y_{2}&z_{1}\!+\!z_{2}\\y_{3}&z_{3}\end{vmatrix}}\!-\!{\vec {j}}{\begin{vmatrix}x_{1}\!+\!x_{2}&z_{1}\!+\!z_{2}\\x_{3}&z_{3}\end{vmatrix}}\!+\!{\vec {k}}{\begin{vmatrix}x_{1}\!+\!x_{2}&y_{1}\!+\!y_{2}\\x_{3}&y_{3}\end{vmatrix}}$ $=\left({\vec {i}}{\begin{vmatrix}y_{1}\!&\!z_{1}\\\!y_{3}&z_{3}\end{vmatrix}}-{\vec {j}}{\begin{vmatrix}x_{1}\!&\!z_{1}\\\!x_{3}&z_{3}\end{vmatrix}}+{\vec {k}}{\begin{vmatrix}x_{1}\!&\!y_{1}\\\!x_{3}&y_{3}\end{vmatrix}}+{\vec {k}}\right)+\left({\vec {i}}{\begin{vmatrix}y_{2}&z_{2}\\y_{3}&z_{3}\end{vmatrix}}-\right.$ $\left.-j{\begin{vmatrix}x_{2}&z_{2}\\x_{3}&z_{3}\end{vmatrix}}+{\vec {k}}{\begin{vmatrix}x_{2}&y_{2}\\x_{3}&y_{3}\end{vmatrix}}\right)$ $={\begin{vmatrix}{\vec {i}}&{\vec {j}}&{\vec {k}}\\x_{1}&y_{1}&z_{1}\\x_{3}&y_{3}&z_{3}\\\end{vmatrix}}+{\begin{vmatrix}{\vec {i}}&{\vec {j}}&{\vec {k}}\\x_{2}&y_{2}&z_{2}\\x_{3}&y_{3}&z_{3}\\\end{vmatrix}}={\vec {a}}\times {\vec {c}}+{\vec {b}}\times {\vec {c}}$. Pra, u vërtetua se $({\vec {a}}+{\vec {b}})\times {\vec {c}}={\vec {a}}\times {\vec {c}}+{\vec {b}}\times {\vec {c}}$.        S h e m b u l l i  17. -  Të njehsohet syprina e paralelogramit të ndërtuar mbi vektorët ${\vec {a}}=2{\vec {i}}+{\vec {j}}-{\vec {k}}$ dhe ${\vec {b}}={\vec {i}}-3{\vec {j}}+4{\vec {k}}$.        Z g j i d h j e : Aplikojmë fomulën (25):        $S={\vec {a}}+{\vec {b}}$ $={\sqrt {{\begin{vmatrix}y_{1}&z_{1}\\y_{2}&z_{2}\end{vmatrix}}^{2}+{\begin{vmatrix}x_{1}&z_{1}\\x_{2}&z_{2}\end{vmatrix}}^{2}+{\begin{vmatrix}x_{1}&y_{1}\\x_{2}&y_{2}\end{vmatrix}}^{2}}}$ $={\sqrt {{\begin{vmatrix}1&-1\\-3&4\end{vmatrix}}^{2}+{\begin{vmatrix}2&-1\\1&4\end{vmatrix}}^{2}+{\begin{vmatrix}2&1\\1&-3\end{vmatrix}}^{2}}}={\sqrt {131}}$.

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