If $\vec{a} = 4\hat{i} + 3\hat{j}$ and $\vec{b} = 2\hat{i} + \lambda\hat{j}$ are parallel vectors,then the value of $\lambda$ is:

$A$ vector $\vec{a}$ has components $2p$ and $1$ with respect to a rectangular Cartesian system. This system is rotated through a certain angle about the origin in the counter-clockwise sense. If,with respect to the new system,$\vec{a}$ has components $p+1$ and $1$,then:

$\vec{c}$ is a unit vector in the direction of the sum of vectors $\vec{a}$ and $\vec{b}$. Where,$\vec{a} = 2 \hat{i} + 2 \hat{j} - 5 \hat{k}$ and $\vec{b} = 2 \hat{i} + \hat{j} + 3 \hat{k}$,then $|\vec{c}| = $ . . . . . . .

Let $A$ be the set of vectors $a = (a_1, a_2, a_3)$ satisfying $\left(\sum_{i=1}^3 \frac{a_i}{2^i}\right)^2 = \sum_{i=1}^3 \frac{a_i^2}{2^i}$. Then,

If $a$ and $b$ are two non-collinear vectors and $xa + yb = 0$,then:

If the magnitude of the sum of two unit vectors is greater than the magnitude of their difference and less than $\sqrt{3}$ times the magnitude of their difference,then the complete set of values for the angle $\theta$ between the vectors is