List of top Mathematics Questions on Vectors in plane and space asked in JEE Main

Let \(\vec{a}_k = (\tan \theta_k) \hat{i} + \hat{j}\) and \(\vec{b}_k = \hat{i} - (\cot \theta_k) \hat{j}\), where \(\theta_k = \frac{2^{k-1}\pi}{2^n+1}\), for some \(n \in \mathbb{N}\), \(n>5\). Then the value of \(\frac{\sum_{k=1}^{n} |\vec{a}_k|^2}{\sum_{k=1}^{n} |\vec{b}_k|^2}\) is ____.
If \(|\vec a|=2\) and \(|\vec b|=3\), then the maximum value of \[ 3\left|\left(\vec a+2\vec b\right)\right| + 4\left|\left(3\vec a-2\vec b\right)\right| \] is:
Let a line pass through two distinct points \( P(-2, -1, 3) \) and \( Q \), and be parallel to the vector \( 3\hat{i} + 2\hat{j} + 2\hat{k} \). If the distance of the point \( Q \) from the point \( R(1, 3, 3) \) is 5, then the square of the area of \( \triangle PQR \) is equal to: