If $\theta$ is the angle between the vectors $\vec{a} = 2\hat{i} + 2\hat{j} - \hat{k}$ and $\vec{b} = 6\hat{i} - 3\hat{j} + 2\hat{k}$,then:

If the vectors $\hat{i}-2x\hat{j}-3y\hat{k}$ and $\hat{i}+3x\hat{j}+2y\hat{k}$ are orthogonal to each other,then the locus of the point $(x, y)$ is

If unit vectors $\bar{a}$ and $\bar{b}$ are perpendicular to each other and a unit vector $\bar{c}$ makes an angle $\theta$ with both $\bar{a}$ and $\bar{b}$,and $\bar{c} = \alpha \bar{a} + \beta \bar{b} + r(\bar{a} \times \bar{b})$,then:

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 $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ are unit vectors such that $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}$,then the value of $3 \overrightarrow{a} \cdot \overrightarrow{b}+2 \overrightarrow{b} \cdot \overrightarrow{c}+\overrightarrow{c} \cdot \overrightarrow{a}$ is

If the scalar projection of the vector $xi - j + k$ on the vector $2i - j + 5k$ is $\frac{1}{\sqrt{30}}$, then the value of $x$ is equal to