If the vectors $a$ and $b$ are mutually perpendicular,then $a \times \{ a \times \{ a \times (a \times b)\} \}$ is equal to

Find all vectors of magnitude $10 \sqrt{3}$ that are perpendicular to the plane of $\hat{i}+2 \hat{j}+\hat{k}$ and $-\hat{i}+3 \hat{j}+4 \hat{k}$.

If $a+2b+3c=0$ and $(a \times b)+(b \times c)+(c \times a)=\lambda(b \times c)$,then the value of $\lambda$ is equal to

If $\vec{a} = 2 \hat{i} + 2 \hat{j} + \hat{k}$,$|\vec{b}| = 6$ and the angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{6}$,then the area of the triangle (in square units) with $\vec{a}$ and $\vec{b}$ as two of its sides 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

The number of vectors of unit length perpendicular to the two vectors $a=(1,1,0)$ and $b=(0,1,1)$ is