Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$,$\vec{b}=2 \hat{i}+4 \hat{j}-5 \hat{k}$ and $\vec{c}=x \hat{i}+2 \hat{j}+3 \hat{k}$,$x \in R$. If $\vec{d}$ is the unit vector in the direction of $\vec{b}+\vec{c}$ such that $\vec{a} \cdot \vec{d}=1$,then $(\vec{a} \times \vec{b}) \cdot \vec{c}$ is equal to

If $\vec{a}, \vec{b}$ and $\vec{c}$ are three non-coplanar vectors and $\vec{p}, \vec{q}$, and $\vec{r}$ are defined by $\vec{p}=\frac{\vec{b} \times \vec{c}}{[\vec{a} \vec{b} \vec{c}]}, \vec{q}=\frac{\vec{c} \times \vec{a}}{[\vec{a} \vec{b} \vec{c}]}, \vec{r}=\frac{\vec{a} \times \vec{b}}{[\vec{a} \vec{b} \vec{c}]}$, then find the value of $(\vec{a}+\vec{b}) \cdot \vec{p} + (\vec{b}+\vec{c}) \cdot \vec{q} + (\vec{c}+\vec{a}) \cdot \vec{r}$.

Let $\vec{a} = (a_1\hat{i} + a_2\hat{j} + a_3\hat{k})$, $\vec{b} = (b_1\hat{i} + b_2\hat{j} + b_3\hat{k})$, and $\vec{c} = (c_1\hat{i} + c_2\hat{j} + c_3\hat{k})$ be three non-zero vectors such that $\vec{a}$ is a unit vector perpendicular to both $\vec{b}$ and $\vec{c}$. If the angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{3}$, then find the value of $\left| \begin{matrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{matrix} \right|^2$.

If $a, b$ and $c$ are three non-coplanar vectors and $p, q$ and $r$ are vectors defined by $p=\frac{b \times c}{[a b c]}, q=\frac{c \times a}{[a b c]}, r=\frac{a \times b}{[a b c]}$,then $(a+b) \cdot p+(b+c) \cdot q+(c+a) \cdot r$ is

If $\overline{a}=2 \hat{\imath}-\hat{\jmath}+\hat{k}$,$\overline{b}=\hat{\imath}+2 \hat{\jmath}-3 \hat{k}$ and $\overline{c}=3 \hat{\imath}+\lambda \hat{\jmath}+5 \hat{k}$ are coplanar,then $\lambda$ is the root of the equation

If three non-zero vectors are $a = a_1 i + a_2 j + a_3 k,$ $b = b_1 i + b_2 j + b_3 k$ and $c = c_1 i + c_2 j + c_3 k.$ If $c$ is the unit vector perpendicular to the vectors $a$ and $b$ and the angle between $a$ and $b$ is $\frac{\pi}{6},$ then $\left| \begin{array}{ccc} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{array} \right|^2$ is equal to