Let $\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{c}=\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$, $\mathbf{d}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$ be four vectors and let $l=\mathbf{b} \cdot \mathbf{c}$ and $m=\mathbf{c} \cdot \mathbf{a}$. Then, $[m \mathbf{b}+l \mathbf{a} \mathbf{b d}]=$

If $\mathbf{a , b , c}$ are three independent vectors and there exists a non zero scalar traid $(l, m, n)$ such that $l(3 \mathbf{a}+2 \mathbf{b}+\mathbf{c})+m(2 \mathbf{a}+2 \mathbf{b}+3 \mathbf{c})+n(\mathbf{a}+2 \mathbf{b}+5 \mathbf{c})=\mathbf{0}$, then

If $\mathbf{a}$ and $\mathbf{b}$ represent two non collinear vectors, the equation $\mathbf{r}=t \mathbf{a}+(1-t) \mathbf{b}$ represents

Let $\mathbf{a , b , c}$ be three vectors such that the magnitude of $\mathbf{b}$ is twice that of $\mathbf{a}$ and magnitude of $\mathbf{c}$ is three times that of $\mathbf{a}$. If the angle between each pair of vectors is $\frac{\pi}{3}$ and $|\mathbf{a}+\mathbf{b}+\mathbf{c}|=5$, then $|\mathbf{c}|+|\mathbf{a}|+|\mathbf{b}|=$