Let $\bar{a}, \bar{b}, \bar{c}$ be vectors such that $\bar{a} \neq \bar{o}, \bar{b} \neq \bar{o}, \bar{a} \times \bar{c} = \bar{b}$ and $\bar{b} \times \bar{c} = \bar{a}$. Then:

Given $a = i + j - k$,$b = -i + 2j + k$,and $c = -i + 2j - k$. $A$ unit vector perpendicular to both $a + b$ and $b + c$ is

Given $|\vec{a}|=\sqrt{3}$,$|\vec{b}|=5$,$\vec{b} \cdot \vec{c}=10$ and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{3}$. If $\vec{a}$ is perpendicular to $\vec{b} \times \vec{c}$,then the value of $|\vec{a} \times(\vec{b} \times \vec{c})|$ is

$A$ non-zero vector $a$ is parallel to the line of intersection of the plane determined by the vectors $i, i + j$ and the plane determined by the vectors $i - j, i + k.$ The angle between $a$ and the vector $i - 2j + 2k$ is

If $u = 2i + 2j - k$ and $v = 6i - 3j + 2k,$ then a unit vector perpendicular to both $u$ and $v$ is

Let the position vectors of two points $P$ and $Q$ be $3 \hat{i} - \hat{j} + 2 \hat{k}$ and $\hat{i} + 2 \hat{j} - 4 \hat{k}$ respectively. Let $R$ and $S$ be two points such that the direction ratios of lines $PR$ and $QS$ are $(4, -1, 2)$ and $(-2, 1, -2)$ respectively. Let lines $PR$ and $QS$ intersect at $T$. If the vector $\vec{TA}$ is perpendicular to both $\vec{PR}$ and $\vec{QS}$ and the length of vector $\vec{TA}$ is $\sqrt{5}$ units,then the modulus of a position vector of $A$ is