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Mathematics · 2026 · 2 marks
CBSE 2026 · Region 1 · Set 3 · Q24
Using vectors, find the area of ABC with vertices A(1,2,3), B(2, -1,4) and C(4, 5, -1).
Using vectors, find the area of $\displaystyle \triangle \mathrm{ABC}$ with vertices A($\displaystyle 1,2,3$), B($\displaystyle 2$, -$\displaystyle 1,4$) and C($\displaystyle 4$, $\displaystyle 5$, -$\displaystyle 1$).
Marking-scheme solution
$\displaystyle \overrightarrow{AB} = \hat{\imath} - 3\hat{\jmath} + \hat{k}$
$\displaystyle \overrightarrow{AC} = 3\hat{\imath} + 3\hat{\jmath} - 4\hat{k}$
$\displaystyle \overrightarrow{AB} \times \overrightarrow{AC} = \begin{vmatrix} \hat{\imath} & \hat{\jmath} & \hat{k} \\ 1 & -3 & 1 \\ 3 & 3 & -4 \end{vmatrix} = 9\hat{\imath} + 7\hat{\jmath} + 12\hat{k}$
The required area $\displaystyle = \dfrac{1}{2}\left|\overrightarrow{AB} \times \overrightarrow{AC}\right| = \dfrac{1}{2}\sqrt{274}$
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