If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are unit vectors such that $\mathbf{a}$ is perpendicular to both $\mathbf{b}, \mathbf{c}$ and angle between $\mathbf{b}, \mathbf{c}$ is $2 \pi / 3$, then $|a+3 b-4 c|^2=$
Let $\mathbf{a}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ be the position vector of a point $A$. Let $\mathbf{b}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\mathbf{c}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ be two vectors and $\mathbf{r}$ be a vector passing through the point $A(\mathbf{a})$ and parallel to the vector $\mathbf{b}$. If the projection of $\mathbf{r}$ on $\mathbf{c}$ is $\frac{9}{\sqrt{6}}$, then $|\mathbf{r}|=$
If $S$ is the circumcentre, $O$ is the orthocentre and $G$ is the centroid of a $\triangle A B C$, then match the items of the List-I with those of the items of List-II given below.
| List-I | List-II | ||
| (i) | (a) | 2 OS | |
| (ii) | (b) | ||
| (iii) | (c) | O | |
| (iv) | OG | (d) | SO |
| (e) | OS |
Then, the correct match is
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