Vector Projection Magnitude Squared Sum (Jee Ma…

Vector Algebra and Components

Medium

Vector Projection Magnitude Squared Sum

Consider two vectors $\vec u = 3\hat i - \hat j$ and $\vec v = 2\hat i + \hat j - \lambda \hat k$ , λ > 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 $\vec u$ and ${\vec v_2}$ is perpendicular to $\vec u$ . Then the value ${\left| {{{\vec v}_1}} \right|^2} + {\left| {{{\vec v}_2}} \right|^2}$ is equal to

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