Hipi Zhdripi i Matematikës/1157

tani marrim shumën e tyre: $({\vec {a}}\cdot {\vec {b}})^{2}+({\vec {a}}\times {\vec {b}})^{2}={\vec {a}}^{2}b^{2}\left[\cos ^{2}({\vec {a}},{\vec {b}})+\sin ^{2}({\vec {a}},{\vec {b}})\right]=a^{2}b^{2}$, dhe vijmë deri të formula e paraqitur.        S h e m b u l l i  16. -  Të thjeshtësohet shprehja $({\vec {a}}+{\vec {b}}-{\vec {c}})\times ({\vec {a}}+{\vec {b}}+{\vec {c}})$.        Z g j i d h j e : I kryejmë veprimet e shënuara: $({\vec {a}}\!+\!{\vec {b}}\!-\!{\vec {c}})\!\times \!({\vec {a}}\!+\!{\vec {b}}\!+\!{\vec {c}})\!=\!{\vec {a}}\!\times \!({\vec {a}}\!+\!{\vec {b}}\!+\!{\vec {c}})\!+\!{\vec {b}}\!\times \!({\vec {a}}+{\vec {b}}\!+\!{\vec {c}})\!-\!{\vec {c}}\!\times \!({\vec {a}}\!+\!{\vec {b}}\!+\!{\vec {c}})\!=$ $=\ldots =2({\vec {a}}\times {\vec {c}}+{\vec {b}}\times {\vec {c}})=2({\vec {a}}+{\vec {b}})\times {\vec {c}}$. 4.2.1. PRODHIMI VEKTORIAL I VEKTORËVE TË SHPREHUR ME KOORDINATA         Në bazë të përkufizimit 4.2.1. përpilohet tabela e prodhimit vektorial të orteve ${\vec {i}},{\vec {j}},{\vec {k}}$: ${\begin{array}{c|c c c}\times &{\vec {i}}&{\vec {j}}&{\vec {k}}\\\hline {\vec {i}}&0&{\vec {k}}&-{\vec {j}}\\{\vec {j}}&-{\vec {k}}&0&{\vec {i}}\\{\vec {k}}&{\vec {j}}&-{\vec {i}}&0\\\end{array}}$ (...24a) ku, për shembull, ${\vec {i}}\times {\vec {k}}={\vec {j}}$, sepse ${\vec {i}},{\vec {k}},-{\vec {j}}$formojnë reperin e djathtë të orteve, ndërkaq .${\vec {j}}\times .{\vec {j}}=0$, sepse $\sin({\vec {j}},{\vec {j}})=\sin 0=0$. Pra, prodhimi vektorial i vektorëve ${\vec {a}}(x_{1},y_{1},z_{1}),{\vec {b}}(x_{2},y2,z_{2})$ përcaktohet me formulën:         ${\vec {a}}\times {\vec {b}}$ $=(x_{1}{\vec {i}}+y{\vec {j}}+z_{1}{\vec {k}})\times (x_{2}{\vec {i}}+y_{2}{\vec {j}}+z_{2}{\vec {k}})$ $=(y_{1}z_{2}-y_{2}a_{1}){\vec {i}}-(x_{1}z_{2}-x_{2}z_{1}){\vec {j}}+(x_{1}y_{2}-x_{2}y_{1}){\vec {k}}$ $={\vec {i}}{\begin{vmatrix}y_{1}&z_{1}\\y_{2}&z_{2}\end{vmatrix}}-{\vec {j}}{\begin{vmatrix}x_{1}&z_{1}\\x_{2}&z_{2}\end{vmatrix}}+{\vec {k}}{\begin{vmatrix}x_{1}&y_{1}\\x_{2}&y_{2}\end{vmatrix}}$

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