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Mathematics · 2024 · 2 marks
CBSE 2024 · Region 1 · Set 1 · Q25
In the given figure, ABCD is a parallelogram. If AB =2 i -4 j +5 k and D B =3 i -6 j +2 k, then find A D and hence find the area of parallelogram…
In the given figure, ABCD is a parallelogram. If $\displaystyle \overrightarrow{\mathrm{AB}}=2 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}$ and $\displaystyle \overrightarrow{D B}=3 \hat{\mathrm{i}}-6 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$, then find $\displaystyle \overrightarrow{A D}$ and hence find the area of parallelogram ABCD .
Marking-scheme solution
$$\begin{aligned}
& \overrightarrow{A D}+\overrightarrow{D B}=\overrightarrow{A B} \\
& \overrightarrow{A D}=(2 \widehat{\imath}-4 \widehat{\jmath}+5 \widehat{\mathrm{k}})-(3 \widehat{\imath}-6 \widehat{\jmath}+2 \widehat{\mathrm{k}}) \\
& =-\widehat{\imath}+2 \widehat{\jmath}+3 \widehat{\mathrm{k}} \\
& \overrightarrow{A D} \times \overrightarrow{A B}=\left|\begin{array}{ccc}
\widehat{\imath} & \widehat{\jmath} & \widehat{\mathrm{k}} \\
-1 & 2 & 3 \\
2 & -4 & 5
\end{array}\right|=22 \widehat{\imath}+11 \widehat{\jmath} \\
& \begin{array}{c}
\text { Area }=|\overrightarrow{A D} \times \overrightarrow{A B}|=|22 \widehat{\imath}+11 \widehat{\jmath}| \\
=\sqrt{605} \text { or } 11 \sqrt{5}
\end{array}
\end{aligned}
$$
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