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Mathematics · 2026 · 2 marks
CBSE 2026 · Region 4 · Set 3 · Q24
If in a parallelogram PQRS, PQ =2 i +3 j + k and QR = i -2 j + k, then find the unit vectors parallel to the diagonals PR and SQ.
If in a parallelogram $\displaystyle \mathrm{PQRS}, \mathrm{PQ}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\displaystyle \overrightarrow{\mathrm{QR}}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$, then find the unit vectors parallel to the diagonals $\displaystyle \overrightarrow{\mathrm{PR}}$ and $\displaystyle \overrightarrow{\mathrm{SQ}}$.
Marking-scheme solution
$\displaystyle \overrightarrow{PR}=\overrightarrow{PQ}+\overrightarrow{QR}=3 \hat{i}+\hat{j}+2 \hat{k}$
So, unit vector parallel to $\displaystyle \overrightarrow{PR}=\dfrac{3}{\sqrt{14}} \hat{i}+\dfrac{1}{\sqrt{14}} \hat{j}+\dfrac{2}{\sqrt{14}} \hat{k}$
Now, $\displaystyle \overrightarrow{SQ}=\overrightarrow{PQ}+\overrightarrow{SP}=\overrightarrow{PQ}-\overrightarrow{PS}=\overrightarrow{PQ}-\overrightarrow{QR}=\hat{i}+5 \hat{j}$
Thus, unit vector parallel to $\displaystyle \overrightarrow{SQ}=\dfrac{1}{\sqrt{26}} \hat{i}+\dfrac{5}{\sqrt{26}} \hat{j}$
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