Let $\vec{a}$ and $\vec{b}$ be two non-collinear vectors of unit modulus. If $\vec{u}=\vec{a}-(\vec{a} \cdot \vec{b}) \vec{b}$ and $\vec{v}=\vec{a} \times \vec{b}$,then $|\vec{v}|=$

Let $\vec{a}=2 \hat{i}+\hat{j}-2 \hat{k}$ and $\vec{b}=\hat{i}+\hat{j} .$ If $\vec{c}$ is a vector such that $\vec{a} \cdot \vec{c}=|\vec{c}|, |\vec{c}-\vec{a}|=2 \sqrt{2}$ and the angle between $(\vec{a} \times \vec{b})$ and $\vec{c}$ is $\frac{\pi}{6}$,then the value of $|(\vec{a} \times \vec{b}) \times \vec{c}|$ is:

$(a - b) \times (a + b) = $

If $a = i - j$,$b = i + j$,$c = i + 3j + 5k$ and $n$ is a unit vector such that $b \cdot n = 0$ and $a \cdot n = 0$,then the value of $|c \cdot n|$ is equal to

Let $\vec a = 2\hat i + \hat j - 2\hat k$ and $\vec b = \hat i + \hat j$. If $\vec c$ is a vector such that $\vec a \cdot \vec c + 2|\vec c| = 0$ and $|\vec c - \vec a| = \sqrt{14}$,and the angle between $\vec a \times \vec b$ and $\vec c$ is $30^o$,then $|(\vec a \times \vec b) \times \vec c|$ is:

Let $\vec{\alpha}, \vec{\beta}, \vec{\gamma}$ be three unit vectors such that $\vec{\alpha} \cdot \vec{\beta} = \vec{\alpha} \cdot \vec{\gamma} = 0$ and the angle between $\vec{\beta}$ and $\vec{\gamma}$ is $30^{\circ}$. Then $\vec{\alpha}$ is