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Mathematics · 2026 · 2 marks
CBSE 2026 · Region 3 · Set 1 · Q24
If for two unit vectors a and b,| a +2 b |=|2 a - b |, then find the angle between a and b.
If for two unit vectors $\displaystyle \overrightarrow{\mathrm{a}}$ and $\displaystyle \overrightarrow{\mathrm{b}},|\overrightarrow{\mathrm{a}}+2 \overrightarrow{\mathrm{~b}}|=|2 \overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|$, then find the angle between $\displaystyle \overrightarrow{\mathrm{a}}$ and $\displaystyle \overrightarrow{\mathrm{b}}$.
Marking-scheme solution
Given that $\displaystyle |\vec{a}|=|\vec{b}|=1$
$\displaystyle |\vec{a}+2\vec{b}|=|2\vec{a}-\vec{b}| \Rightarrow |\vec{a}+2\vec{b}|^{2}=|2\vec{a}-\vec{b}|^{2}$
$\displaystyle \Rightarrow |\vec{a}|^{2}+4|\vec{b}|^{2}+4\vec{a} \cdot \vec{b}=4|\vec{a}|^{2}+|\vec{b}|^{2}-4\vec{a} \cdot \vec{b} \Rightarrow \vec{a} \cdot \vec{b}=0$
$\displaystyle \Rightarrow$ Angle between $\displaystyle \vec{a}$ and $\displaystyle \vec{b}=\dfrac{\pi}{2}$
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