Question
Download Solution PDFयदि \(\overrightarrow{AC}=2\hat{i}+\hat{j}+\hat{k}\) और \(\overrightarrow{BD}=-\hat{i}+3\hat{j}+2\hat{k}\) है, तो चतुर्भुज ABCD का क्षेत्रफल कितना होगा?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFसंकल्पना:
चतुर्भुज ABCD का क्षेत्रफल = \(\rm \dfrac 12 |\vec {AC} \times \vec {BD}|\) , जहां \(\rm \vec {AC} \;\; \text {and }\;\; \vec {BD}\) विकर्ण हैं।
गणना:
माना\(\overrightarrow{AC}=2\hat{i}+\hat{j}+\hat{k}\) और \(\overrightarrow{BD}=-\hat{i}+3\hat{j}+2\hat{k}\) चतुर्भुज ABCD के विकर्ण हैं।
चतुर्भुज ABCD का क्षेत्रफल = \(\rm \dfrac 12 |\vec {AC} \times \vec {BD}|\) ....(1)
⇒ \(\rm (\vec {AC} \times \vec {BD}) =\) \(\begin{vmatrix} \vec i&\vec j & \vec k \\ 2&1 &1 \\ -1&3 &2 \end{vmatrix}\)
⇒ \(\rm (\vec {AC} \times \vec {BD}) =\) \(\rm -\vec i - 5\vec j + 7 \vec k\)
⇒ \(\rm |\vec {AC} \times \vec {BD}| = \sqrt {(-1)^2+(-5)^2+(7)^2}\)
⇒ \(\rm |\vec {AC} \times \vec {BD}| = \sqrt {75} = 5 \sqrt 3\)
समीकरण (1) से, हमारे पास है
चतुर्भुज ABCD का क्षेत्रफल = \(\rm \dfrac{5\sqrt3}{2}\)
Last updated on Jun 12, 2025
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