Question

Select the correct answer
The area of the triangle formed by origin O and the two points A and B given by à = i + 25 + 3k and b= -31 - 2j + k respectively is,

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{In\;\triangle\;OAB,}$

$\mathsf{\overrightarrow{OA}=\overrightarrow{a}=\overrightarrow{i}+2\overrightarrow{j}+3\overrightarrow{K}}$

$\mathsf{\overrightarrow{OB}=\overrightarrow{b}=-3\overrightarrow{i}-2\overrightarrow{j}+\overrightarrow{K}}$

$\underline{\textbf{To find:}}$

$\textsf{Area of triangle OAB}$

$\underline{\textbf{Solution:}}$

$\underline{\textsf{Concept used:}}$

$\boxed{\mathsf{Area\;of\;triangle\;having\;two\;sides\;\overrightarrow{a}\;and\;\overrightarrow{b}=\dfrac{1}{2}|\overrightarrow{a}{\times}\overrightarrow{b}|}}$

$\mathsf{\overrightarrow{a}{\times}\overrightarrow{b}}$

$\mathsf{=\left|\begin{array}{ccc}\overrightarrow{i}&\overrightarrow{j}&\overrightarrow{k}\\1&2&3\\-3&-2&1\end{array}\right|}$

$\mathsf{=(2+6)\overrightarrow{i}-(1+9)\overrightarrow{j}+(-2+6)\overrightarrow{k}}$

$\mathsf{=8\overrightarrow{i}-10\overrightarrow{j}+4\overrightarrow{k}}$

$\mathsf{=2(4\overrightarrow{i}-5\overrightarrow{j}+2\overrightarrow{k})}$

$\implies\mathsf{|\overrightarrow{a}{\times}\overrightarrow{b}|=2\sqrt{4^2+(-5)^2+2^2}}$

$\implies\mathsf{|\overrightarrow{a}{\times}\overrightarrow{b}|=2\sqrt{16+25+4}}$

$\implies\mathsf{|\overrightarrow{a}{\times}\overrightarrow{b}|=2\sqrt{45}}$

$\implies\mathsf{|\overrightarrow{a}{\times}\overrightarrow{b}|=6\sqrt{5}}$

$\mathsf{Now,}$

$\underrline{\textsf{Area of triangle OAB}}$

$\mathsf{=\dfrac{1}{2}|\overrightarrow{a}{\times}\overrightarrow{b}|}$

$\mathsf{=\dfrac{1}{2}(6\sqrt{5})}$

$\mathsf{=3\sqrt{5}\;square\;units}$

Answer 2

Answer:

3√5

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