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Mathematics · 2025 · 2 marks
CBSE 2025 · Region 2 · Set 1 · Q21
(a) A vector a makes equal angles with all the three axes. If the magnitude of the vector is 5 √ 3 units, then find a.
(a)
A vector $\displaystyle \vec{\mathrm{a}}$ makes equal angles with all the three axes. If the magnitude of the vector is $\displaystyle 5 \sqrt{3}$ units, then find $\displaystyle \vec{\mathrm{a}}$.
OR
(b)
If $\displaystyle \vec{\alpha}$ and $\displaystyle \vec{\beta}$ are position vectors of two points $\displaystyle \mathrm{P}$ and $\displaystyle \mathrm{Q}$ respectively, then find the position vector of a point R in QP produced such that $\displaystyle \mathrm{QR}=\frac{3}{2} \mathrm{QP}$.
Marking-scheme solution
Let $\displaystyle \alpha$ be the angle which the vector $\displaystyle \overrightarrow{\mathrm{a}}$ makes with all the three axes.
Then $\displaystyle 3 \cos ^{2} \alpha=1$
\[\Rightarrow \cos \alpha=\frac{1}{\sqrt{3}}
\]
The unit vector along the vector $\displaystyle \vec{\mathrm{a}}=\frac{1}{\sqrt{3}}(\hat{\imath}+\hat{\jmath}+\hat{k})$
\[\vec{\mathrm{a}}=5(\hat{\imath}+\hat{\jmath}+\hat{k})
\]
\[\begin{array}{lll}
\mathrm{R}(\overrightarrow{\mathrm{x}}) & \mathrm{P}(\vec{\alpha}) & \mathrm{Q}(\vec{\beta}) \\
\frac{\mathrm{QR}}{\mathrm{QP}}=\frac{3}{2} &
\end{array}
\]
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