If $a = i + j + k$,$b = i + j$,$c = i$ and $(a \times b) \times c = \lambda a + \mu b$,then $\lambda + \mu = \dots$

If $a, b, c$ are three unit vectors such that $a \times (b \times c) = \frac{\sqrt{3}}{2} b + \frac{1}{2} c$,then the angles between $a, b$ and $a, c$ respectively are

If $a, b$ and $c$ are three vectors with magnitudes $1, 1$ and $2$ respectively and $a \times (a \times c) + b = 0$, then the angle between $a$ and $c$ is

Let $\vec{p}, \vec{q},$ and $\vec{r}$ be three non-coplanar unit vectors equally inclined to each other at an acute angle $\theta$. The value of $|\vec{p} \times (\vec{q} \times \vec{r})|$ is:

If $a = i + j - k$,$b = i - j + k$,and $c = i - j - k$,then $a \times (b \times c)$ is:

Let $\vec{a}=\hat{i}+\hat{j}+2 \hat{k}$ and $\vec{b}=-\hat{i}+2 \hat{j}+3 \hat{k}$. Then the vector product $(\vec{a}+\vec{b}) \times((\vec{a} \times((\vec{a}-\vec{b}) \times \vec{b})) \times \vec{b})$ is equal to: