If the vectors $\vec{a} + \lambda \vec{b} + 3\vec{c}$,$-2\vec{a} + 3\vec{b} - 4\vec{c}$,and $\vec{a} - 3\vec{b} + 5\vec{c}$ are coplanar,and $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar,find the value of $\lambda$.

Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$,$\vec{b}$,and $\vec{c} = \hat{j} - \hat{k}$ be three vectors such that $\vec{a} \times \vec{b} = \vec{c}$ and $\vec{a} \cdot \vec{b} = 1$. If the length of the projection vector of the vector $\vec{b}$ on the vector $\vec{a} \times \vec{c}$ is $l$,then the value of $3l^{2}$ is equal to $.....$

$a, b, c$ are three non-zero,non-coplanar vectors and $p, q, r$ are three other vectors such that $p = \frac{b \times c}{a \cdot (b \times c)}$,$q = \frac{c \times a}{a \cdot (b \times c)}$,$r = \frac{a \times b}{a \cdot (b \times c)}$. Then $[p, q, r]$ equals

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

If the vectors $i + 3j$,$5k$,and $Pi - j$ are coplanar,find the value of $P$.

If $a = i - j + k$,$b = i + 2j - k$ and $c = 3i + pj + 5k$ are coplanar,then the value of $p$ is: