Orthogonal System of Vectors

Q. The unit vector in the direction of sum of the vectors $\overrightarrow{A}=2\hat{i}+4\hat{j}-7\hat{k}$ and $\overrightarrow{B}=4\hat{i}-6\hat{j}+4\hat{k}$ will be

1. $6\hat{i}-2\hat{j}-3\hat{k}$
2. $\hat{i}-\hat{j}-\hat{k}$
3. $\frac{6\hat{i}-2\hat{j}-3\hat{k}}{7}$

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