Orthogonal System of Vectors

Q. Let $\overrightarrow{p},\overrightarrow{q},\overrightarrow{r}$ be three mutually perpendicular vectors of the same magnitude. If a vector $\overrightarrow{x}$ satisfies the equation
$\overrightarrow{p}\left[\right(\overrightarrow{x}-\overrightarrow{q})\times \overrightarrow{p}]+\overrightarrow{q}\times \left[\right(\overrightarrow{x}-\overrightarrow{r})\times \overrightarrow{q}]+\overrightarrow{r}\times \left[\right(\overrightarrow{x}-\overrightarrow{p})\times \overrightarrow{r}]=\overrightarrow{0}$, then $\overrightarrow{x}$ is given by

1. $\frac{1}{2}(\overrightarrow{p}+\overrightarrow{q}-2\overrightarrow{r})$
2. $\frac{1}{2}(\overrightarrow{p}+\overrightarrow{q}+\overrightarrow{r})$
3. $\frac{1}{3}(\overrightarrow{p}+\overrightarrow{q}+\overrightarrow{r})$
4. $\frac{1}{3}(2\overrightarrow{p}+\overrightarrow{q}-\overrightarrow{r})$

Q. If $\hat{i},\hat{j}$ and $\hat{k}$ are the unit vectors along positive direction of x, y and z axes respectively and $\overrightarrow{r}=a\hat{i}+b\hat{j}+c\hat{k}$ then the component of $\overrightarrow{r}$ along z axis is b.

1. True
2. False

Q. Let $\overrightarrow{p},\overrightarrow{q},\overrightarrow{r}$ be three mutually perpendicular vectors of the same magnitude. If a vector $\overrightarrow{x}$ satisfies the equation
$\overrightarrow{p}\left[\right(\overrightarrow{x}-\overrightarrow{q})\times \overrightarrow{p}]+\overrightarrow{q}\times \left[\right(\overrightarrow{x}-\overrightarrow{r})\times \overrightarrow{q}]+\overrightarrow{r}\times \left[\right(\overrightarrow{x}-\overrightarrow{p})\times \overrightarrow{r}]=\overrightarrow{0}$, then $\overrightarrow{x}$ is given by

1. $\frac{1}{2}(\overrightarrow{p}+\overrightarrow{q}-2\overrightarrow{r})$
2. $\frac{1}{2}(\overrightarrow{p}+\overrightarrow{q}+\overrightarrow{r})$
3. $\frac{1}{3}(\overrightarrow{p}+\overrightarrow{q}+\overrightarrow{r})$
4. $\frac{1}{3}(2\overrightarrow{p}+\overrightarrow{q}-\overrightarrow{r})$