List of top Mathematics Questions on Addition of Vectors asked in JEE Advanced

Let \(\vec{a}, \vec{b}\) be two vectors, and let \(P, Q\) and \(R\) be the points with position vectors \(\vec{a}, \vec{b}\) and \(\vec{a} + \vec{b}\), respectively, with respect to the origin \(O\). If \[ |\vec{a} + \vec{b}| = \sqrt{21}, \qquad |\vec{a} - \vec{b}| = 3, \] and \(\vec{a}\) and \((\vec{a} - \vec{b})\) are perpendicular to each other, then the area of the triangle \(OPR\) is:
For real numbers \(\alpha,\beta,\gamma,\delta\) and \(\mu\), consider the matrix \[ M= \begin{bmatrix} \alpha & \frac1{\sqrt2} & -\frac1{\sqrt2} \frac1{\sqrt3} & \beta & \frac1{\sqrt3} \gamma & \delta & \mu \end{bmatrix} \] Suppose that \[ MM^T=I \] where \(M^T\) is the transpose of the matrix \(M\) and \(I\) is the \(3\times3\) identity matrix. Let \[ \vec u=\alpha\hat i+\frac1{\sqrt3}\hat j+\gamma\hat k \] \[ \vec v=\frac1{\sqrt2}\hat i+\beta\hat j+\delta\hat k \] \[ \vec w=-\frac1{\sqrt2}\hat i+\frac1{\sqrt3}\hat j+\mu\hat k \] Match each entry in List-I to the correct entry in List-II and choose the correct option.