ভেক্টর জ্যামিতি গানিতীক সমাধান- ৯

৪১) দেখাও যে, \displaystyle [ \ \overline{l} \ \ \overline{m} \ \ \overline{n} \ ] \ \ [ \ \overline{a} \ \ \overline{b} \ \ \overline{c} \ ] = \displaystyle \begin{array}{{>{\displaystyle}l}}\ \ \ \ \ \ \ \ \ \ \overline{l} .\overline{a} \ \ \ \ \overline{l} .\overline{b} \ \ \ \ \overline{l} .\overline{c}\\overline{m} .\overline{a} \ \ \ \overline{m} .\overline{b} \ \ \ \ \overline{m} .\overline{c} \ \\overline{n} .\overline{a} \ \ \ \ \overline{n} .\overline{b} \ \ \ \ \ \overline{n} .\overline{c}\end{array}
যদি \displaystyle \overline{l} =\overline{a} ,\ \overline{m} =\overline{b} ,\ \overline{n} =\overline{c} হয়, তবে প্রমান কর যে, \displaystyle [ \ \overline{l} \ \overline{m} \ \overline{n} \ ] \ [ \ \overline{a} \ \overline{b} \ \overline{c} \ ] =[ \ \overline{a} \ \overline{b} \ \overline{c} \ ]^{2}

৪২) দেখাও যে, \displaystyle [ \ \overline{l} \ \ \overline{m} \ \ \overline{n} \ ] \ \ ( \ \overline{a} \ X\ \overline{b} \ ) =\displaystyle \begin{array}{{>{\displaystyle}l}}\ \overline{l} \ \ \ \ \overline{l} .\overline{a} \ \ \ \ \overline{l} .\overline{b}\\ \ \ \ \ \ \ \ \overline{m} \ \ \ \ \ \ \overline{m} .\overline{a} \ \ \ \ \ \ \ \overline{m} .\overline{b} \ \\ \ \ \ \ \ \ \ \overline{n} . \ \ \ \ \ \ \overline{n} .\overline{a} \ \ \ \ \ \ \ \ \overline{n} .\overline{b}\end{array}

৪৩) \displaystyle \hat{a} এবং \displaystyle \hat{b} একক ভেক্টরদ্বয়ের মধ্যবর্তী কোণ \displaystyle \theta হলে দেখাও যে, \displaystyle sin\frac{\theta }{2} =\frac{1}{2}| \hat{a} -\hat{b}|

৪৩) \displaystyle \hat{a} এবং \displaystyle \hat{b} একক ভেক্টরদ্বয়ের মধ্যবর্তী কোণ \displaystyle \theta হলে দেখাও যে, \displaystyle sin\frac{\theta }{2} =\frac{1}{2}| \hat{a} -\hat{b}|
৪৪) যদি \displaystyle \overline{a} =a_{1}\hat{i} +a_{2}\hat{j} +a_{3}\hat{k} ,\ \overline{b} =b_{1}\hat{i} +b_{2}\hat{j} +b_{3}\hat{k} ,\ \overline{c} =c_{1}\hat{i} +c_{2}\hat{j} +c_{3}\hat{k} , হয় , তবে দেখাও যে, \displaystyle [ \ \overline{a} \ \overline{b} \ \overline{c} \ ] =a_{1}( b_{2} c_{3} -b_{3} c_{2}) +a_{2}( b_{3} c_{1} -b_{1} c_{3}) +a_{3}( b_{1} c_{2} -b_{2} c_{1})

৪৫)\displaystyle \overline{a} =a_{1}\hat{i} +a_{2}\hat{j} +a_{3}\hat{k} ,\ \overline{b} =b_{1}\hat{i} +b_{2}\hat{j} +b_{3}\hat{k} ,\ \overline{c} =c_{1}\hat{i} +c_{2}\hat{j} +c_{3}\hat{k} , হলে , প্রমাণ কর যে, \displaystyle \overline{a} .(\overline{b} X\overline{c}) =\overline{b} .( \ \overline{c} X\overline{a} \ ) =\overline{c} .( \ \overline{a} X\overline{b} \ )

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