Given the following simultaneous equations for vectors $x$ and $y$:
$(i) x + y = a$
$(ii) x \times y = b$
$(iii) x \cdot a = 1$
Then $x = ?, y = ?$

If $|\vec{a}|=3, |\vec{b}|=5$ and $|\vec{c}|=7$ and $\vec{a}+\vec{b}+\vec{c}=\vec{0}$,then the angle between $\vec{a}$ and $\vec{b}$ is

In quadrilateral $ABCD$,$AB=a$,$BC=b$,$AD=b-a$. If $M$ is the midpoint of $BC$ and $N$ is a point on $DM$ such that $DN=\left(\frac{4}{5}\right) DM$,then $5 AN=$

Let $S$ be the set of all $a \in \mathbb{R}$ for which the angle between the vectors $\vec{u} = a(\log_{e} b) \hat{i} - 6 \hat{j} + 3 \hat{k}$ and $\vec{v} = (\log_{e} b) \hat{i} + 2 \hat{j} + 2a(\log_{e} b) \hat{k}$ is acute,where $b > 1$. Then $S$ is equal to:

If $a = 2i + j + 2k$ and $b = 5i - 3j + k$,then the projection of $b$ on $a$ is

If $a, b, c$ are unit vectors such that $a + b + c = 0,$ then $a \cdot b + b \cdot c + c \cdot a = $