The projection of the line segment joining the points $P(1, -1, 3)$ and $Q(2, -4, 11)$ on the line joining the points $A(-1, 2, 3)$ and $B(3, -2, 10)$ is

If the angle between the vectors $a$ and $b$ is $\theta$ and $a \cdot b = \cos \theta$,then the true statement is:

Consider the vectors $\vec{a}=3 \hat{i}+5 \hat{j}+2 \hat{k}$, $\vec{b}=2 \hat{i}-3 \hat{j}-5 \hat{k}$ and $\vec{c}=-5 \hat{i}-2 \hat{j}+3 \hat{k}$. If $l, m$ and $n$ are the lengths of the projections of $\vec{a}$ on $\vec{b}$, $\vec{b}$ on $\vec{c}$ and $\vec{c}$ on $\vec{a}$ respectively, then:

If $|a \cdot b| = 3$ and $|a \times b| = 4$,then the angle between $a$ and $b$ is

$A$ force of magnitude $6$ acts along the vector $(9, 6, -2)$ and passes through a point $A(4, -1, -7)$. The moment of the force about the point $O(1, -3, 2)$ is

If $P(6, 10, 10)$,$Q(1, 0, -5)$,$R(6, -10, \lambda)$ are vertices of a triangle right-angled at $Q$,then the value of $\lambda$ is ....