If $a$ makes an acute angle with $b$,$r \cdot a = 0$ and $r \times b = c \times b$,then $r=$

In $\triangle ABC$,if $S$ is the circumcentre and $O$ is the orthocentre,then $\vec{OA} + \vec{OB} + \vec{OC} = $

Let $\overrightarrow{a} = 2\hat{i} - 3\hat{j} + 4\hat{k}$ and $\overrightarrow{b} = 7\hat{i} + \hat{j} - 6\hat{k}$. If $\overrightarrow{r} \times \overrightarrow{a} = \overrightarrow{r} \times \overrightarrow{b}$ and $\overrightarrow{r} \cdot (\hat{i} + 2\hat{j} + \hat{k}) = -3$,then $\overrightarrow{r} \cdot (2\hat{i} - 3\hat{j} + \hat{k})$ is equal to:

If $a$ and $b$ are two unit vectors and $\theta$ is the angle between them,then the unit vector along the angular bisector of $a$ and $b$ is given by

Find the torque of the couple formed by forces $(9, -1, 2)$ and $(3, -2, 1)$ acting at the points $5\hat{i} + \hat{k}$ and $-5\hat{i} - \hat{k}$ respectively.

If $\hat{i}, \hat{j}, \hat{k}$ are unit vectors along the positive direction of $x, y$ and $z$-axes,then which of the following is a false statement?