جواب کار در کلاس صفحه 80 درس 3 هندسه دوازدهم (بردارها)

\(\begin{array}{l}\left. \begin{array}{l}\overrightarrow i = (1\;,\;0\;,\;0)\;,\;\overrightarrow j = (0\;,\;1\;,\;0)\\\\\overrightarrow i \cdot \overrightarrow j = 1 \times 0 + 0 \times 1 + 0 \times 0 = 0\;\\\\{\left| {\overrightarrow j } \right|^2} = {0^2} + {1^2} + {0^2} = 1\end{array} \right\}\\\\ \Rightarrow \overrightarrow {i'} = \frac{{\overrightarrow i \cdot \overrightarrow j }}{{{{\left| {\overrightarrow j } \right|}^2}}}\overrightarrow j = \frac{0}{1}(0\;,\;1\;,\;0) = (0\;,\;0\;,\;0) = \overrightarrow O \end{array}\)

فرض کنیم

\(\overrightarrow b = ({b_1}\;,\;{b_2}\;,\;{b_3}) \ne \overrightarrow O \;,\;\overrightarrow a = ({a_1}\;,\;{a_2}\;,\;{a_3}) \ne \overrightarrow O \) و  \(\overrightarrow a \bot \overrightarrow b \) 

در این صورت \(\overrightarrow a \cdot \overrightarrow b = \;\;0\) و تصویر \(\overrightarrow a \) بر \(\overrightarrow b \) برابر است با :

\(\begin{array}{l}\left. \begin{array}{l}\overrightarrow i = (1\;,\;0\;,\;0)\;,\;\overrightarrow j = (0\;,\;1\;,\;0)\\\\\overrightarrow i \cdot \overrightarrow j = 1 \times 0 + 0 \times 1 + 0 \times 0 = 0\;\\\\{\left| {\overrightarrow j } \right|^2} = {0^2} + {1^2} + {0^2} = 1\end{array} \right\}\\\\ \Rightarrow \overrightarrow {a'} = \frac{{\overrightarrow a \cdot \overrightarrow b }}{{{b^2}}}\overrightarrow b = \frac{0}{{{b^2}}}({b_1}\;,\;{b_2}\;,\;{b_3})\\\\ = (0\;,\;0\;,\;0) = \overrightarrow O \end{array}\)

فرض کنیم

\(\overrightarrow b = ({b_1}\;,\;{b_2}\;,\;{b_3}) \ne \overrightarrow O \;,\;\overrightarrow a = ({a_1}\;,\;{a_2}\;,\;{a_3}) \ne \overrightarrow O \) و \(\overrightarrow a = k\overrightarrow b \) 

در این صورت :

\(\overrightarrow a \cdot \overrightarrow b = k\overrightarrow b \cdot \overrightarrow b = k{b^2}\)

 تصویر \(\overrightarrow a \) بر \(\overrightarrow b \) برابر است با :

\(\overrightarrow {a'} = \frac{{\overrightarrow a \cdot \overrightarrow b }}{{{b^2}}}\overrightarrow b = \frac{{k\overrightarrow b \cdot \overrightarrow b }}{{{b^2}}}\overrightarrow b = \frac{{k{b^2}}}{{{b^2}}}\overrightarrow b = k\overrightarrow b = \overrightarrow a \)