The vector projection of a vector $3\hat i + 4\hat k$ on the $Y$-axis is

If $\vec{A} + \vec{B} = \vec{C}$ and $|\vec{A}| + |\vec{B}| = |\vec{C}|$,what can be said about the direction of these vectors?

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.

Write the distributive law for the product of two vectors.

Which of the following quantity/quantities are dependent on the choice of orientation of the co-ordinate axes?
$(a)$ $\vec{a}+\vec{b}$
$(b)$ $3a_x+2b_y$
$(c)$ $(\vec{a}+\vec{b}-\vec{c})$

Find $|\Delta \overrightarrow{a}|$ and $\Delta a$ when a vector $\overrightarrow{a}$ is rotated by an angle $d\theta$.