The scalar product of the vector $\hat{i}+\hat{j}+\hat{k}$ with a unit vector along the sum of the vectors $2 \hat{i}+4 \hat{j}-5 \hat{k}$ and $\lambda \hat{i}+2 \hat{j}+3 \hat{k}$ is equal to $1$. Then the value of $\lambda$ is:

$A$ vector of magnitude $\sqrt{51}$ which makes equal angles with the vectors $\bar{a}=\frac{1}{3}(\bar{i}-2 \bar{j}+2 \bar{k})$,$\bar{b}=\frac{1}{5}(-4 \bar{i}-3 \bar{k})$ and $\bar{c}=\bar{j}$,is

If $|\vec{a}| = 3, |\vec{b}| = 4, |\vec{c}| = 5$ such that each vector is perpendicular to the sum of the other two, then $|\vec{a} + \vec{b} + \vec{c}|$ is equal to

If the position vectors of the vertices $A, B$ and $C$ of a $\Delta ABC$ are respectively $4\hat{i} + 7\hat{j} + 8\hat{k}$,$2\hat{i} + 3\hat{j} + 4\hat{k}$ and $2\hat{i} + 5\hat{j} + 7\hat{k}$,then the position vector of the point,where the bisector of $\angle A$ meets $BC$ is

Three vectors $\vec{a}, \vec{b}$ and $\vec{c}$ satisfy the condition $\vec{a}+\vec{b}+\vec{c}=\vec{0}.$ Evaluate the quantity $\mu=\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a},$ if $|\vec{a}|=1, |\vec{b}|=4$ and $|\vec{c}|=2.$

The projection of the vector $\hat{i} + \hat{j} + \hat{k}$ on the line $\vec{r} = 3\hat{i} - \hat{j} + \lambda(\hat{i} + 2\hat{j} + 3\hat{k})$ is: