The component of a vector $P=3 \hat{i}+4 \hat{j}$ along the direction $(\hat{i}+2 \hat{j})$ is

The angle made by a vector $\vec{B} = 3 \hat{i} + 2 \hat{j} + 4 \hat{k}$ with the $y$-axis is:

The component of a vector $\vec{A}$ along a direction making an angle $\theta$ with it is given by $A \cos \theta$. What can be said about the magnitude of this component?

The five sides of a regular pentagon are represented by vectors $A_1, A_2, A_3, A_4$ and $A_5$ in cyclic order as shown in the figure. Corresponding vertices are represented by vectors $B_1, B_2, B_3, B_4$ and $B_5$,drawn from the center of the pentagon. Then,$B_2 + B_3 + B_4 + B_5$ is equal to:

If a vector $\overrightarrow{P}$ makes angles $\alpha, \beta,$ and $\gamma$ with the $X, Y,$ and $Z$ axes respectively,then $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = $

Given $a+b+c+d=0,$ which of the following statements are correct:
$(a)$ $a, b, c,$ and $d$ must each be a null vector.
$(b)$ The magnitude of $(a+c)$ equals the magnitude of $(b+d)$.
$(c)$ The magnitude of $a$ can never be greater than the sum of the magnitudes of $b, c,$ and $d$.
$(d)$ $(b+c)$ must lie in the plane of $a$ and $d$ if $a$ and $d$ are not collinear,and in the line of $a$ and $d$ if they are collinear.