Consider two vectors $\vec{\mathrm{u}}=3\overset{^}{\mathrm{i}}-\overset{^}{\mathrm{j}}$ and $\vec{\mathrm{v}}=2\overset{^}{\mathrm{i}}+\overset{^}{\mathrm{j}}-\lambda \overset{^}{\mathrm{k}},\lambda 0$. The angle between them is given by ${\cos }^{-1}\left(\frac{\sqrt{5}}{2\sqrt{7}}\right)$. Let $\vec{v}={\vec{v}}_{1}+{\vec{v}}_{2}$, where ${\vec{v}}_{1}$ is parallel to $\vec{\mathrm{u}}$ and ${\vec{\mathrm{v}}}_{2}$ is perpendicular to $\vec{\mathrm{u}}$. Then the value ${\left|{\vec{v}}_{1}\right|}^{2}+{\left|{\vec{v}}_{2}\right|}^{2}$ is equal to [JEE Main 2025, 4 Apr (Shift 1)]

Question

Consider two vectors $\vec{\mathrm{u}}=3\overset{^}{\mathrm{i}}-\overset{^}{\mathrm{j}}$ and $\vec{\mathrm{v}}=2\overset{^}{\mathrm{i}}+\overset{^}{\mathrm{j}}-\lambda \overset{^}{\mathrm{k}},\lambda >0$. The angle between them is given by ${\cos }^{-1}\left(\frac{\sqrt{5}}{2\sqrt{7}}\right)$. Let $\vec{v}={\vec{v}}_{1}+{\vec{v}}_{2}$, where ${\vec{v}}_{1}$ is parallel to $\vec{\mathrm{u}}$ and ${\vec{\mathrm{v}}}_{2}$ is perpendicular to $\vec{\mathrm{u}}$. Then the value ${\left|{\vec{v}}_{1}\right|}^{2}+{\left|{\vec{v}}_{2}\right|}^{2}$ is equal to [JEE Main 2025, 4 Apr (Shift 1)]

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Consider two vectors \(\vec{\mathrm{u}}=3\overset{^}{\mathrm{i}}-\overset{^}{\mathrm{j}}\) and \(\vec{\mathrm{v}}=2\over…

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