The position vector of a point R which divides the line joining the points A(-2, 1, 3) and B(3, 5, -2) internally in the ratio 2:1, is

Let $\vec{a}$ be any vector such that $\vec{a}.\hat{i} = \vec{a}.(\hat{i}+\hat{j}) = \vec{a}.(\hat{i}+\hat{j}+\hat{k}) = 1$, then $\vec{a}$ is

If $\vec{a}$ is a unit vector perpendicular to both the vectors $\vec{b} = \hat{j} + \hat{2k}$ and $\vec{c} = \hat{i} + 2\hat{j}$, then $\hat{a}$ is equal to

Match List-I with List-II

Choose the correct answer from the options given below:

If the points P, Q, R with position vectors $5\hat{i} + \lambda\hat{j}$, $20\hat{i} - \hat{j}$ and $15\hat{i} - 6\hat{j}$ respectively are collinear, then the value of $\lambda$ is

A vector of magnitude 9, which is perpendicular to both the vectors $(4\hat{i} - \hat{j} + 8\hat{k})$ and $(-\hat{j} + \hat{k})$ is

If $\vec a=\hat i+\hat j+2\hat k$ and $\vec b=2\hat i+\hat j+2\hat k$, then the unit vector in the direction of $2\vec a-\vec b$ is equal to :