If $a = i + j - 2k$,then $\sum \{(a \times i) \times j\}^2$ is equal to

If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$,$\vec{b} = \hat{i} + \hat{j}$,$\vec{c} = \hat{i}$ and $(\vec{a} \times \vec{b}) \times \vec{c} = \lambda \vec{a} + \mu \vec{b}$,then $\lambda + \mu$ is equal to :-

If $\vec{a}=2 \hat{i}+3 \hat{j}$,$\vec{b}=3 \hat{j}+4 \hat{k}$,and $\vec{c}=5 \hat{i}+4 \hat{k}$ are three vectors,then a vector which is perpendicular to $\vec{a}$ and $\vec{b} \times \vec{c}$ is

Let $a, b, c$ be three unit vectors such that $a \times(b \times c)=\frac{1}{2} b$. If the angle between $a$ and $b$ is $\theta_1$ and the angle between $a$ and $c$ is $\theta_2$,then $\theta_1+\theta_2$ is equal to (in $^{\circ}$)

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non-zero vectors such that no two of them are collinear and $(\vec{a} \times \vec{b}) \times \vec{c} = \frac{1}{3}|\vec{b}| |\vec{c}| \vec{a}$. If $\theta$ is the angle between vectors $\vec{b}$ and $\vec{c}$,then a value of $\sin \theta$ is:

Let $\vec{a}=2 \hat{i}-\hat{j}+5 \hat{k}$ and $\vec{b}=\alpha \hat{i}+\beta \hat{j}+2 \hat{k}$. If $((\vec{a} \times \vec{b}) \times \hat{i}) \cdot \hat{k}=\frac{23}{2}$,then $|\vec{b} \times 2 \hat{j}|$ is equal to.