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
Let $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be three vectors such that $\mathbf{a} \cdot \mathbf{a}=\mathbf{b} \cdot \mathbf{b}=\mathbf{c} \cdot \mathbf{c}=5$ and $|\mathbf{a}+\mathbf{b}-\mathbf{c}|^2+|\mathbf{b}+\mathbf{c}-\mathbf{a}|^2+|\mathbf{c}+\mathbf{a}-\mathbf{b}|^2=50$, then $\mathbf{a} \cdot \mathbf{b}+\mathbf{b} \cdot \mathbf{c}+\mathbf{c} \cdot \mathbf{a}=$
Let $\mathbf{c}$ be a vector coplanar with the unit vectors $\mathbf{a}, \mathbf{b}$ and let $\mathbf{d}$ be the unit vector perpendicular to $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$. If $[\mathbf{a} \mathbf{b} \mathbf{d}] \mathbf{c}-[\mathbf{a} \mathbf{b} \mathbf{c}] \mathbf{d}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and the angle between $\mathbf{a}$ and $\mathbf{b}$ is $30^{\circ}$, then $|\mathbf{c}|=$
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