If $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are non-coplanar vectors and the points $\lambda \mathbf{a}+3 \mathbf{b}-\mathbf{c}, \mathbf{a}-\lambda \mathbf{b}+3 \mathbf{c}, 3 \mathbf{a}+4 \mathbf{b}-\lambda \mathbf{c}$ and $\mathbf{a}-6 b+6 \mathbf{c}$ are coplanar, then one of the values of $\lambda$ is

If the position vectors of $\mathbf{P}$ and $\mathbf{Q}$ are $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-7 \hat{\mathbf{k}}$ and $5 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ respectively, then the cosine of the angle between $P Q$ and $Z$-axis is

$\mathbf{a}, \mathbf{b}, \mathbf{c}$ are three-unit yectors such that $|\mathbf{a}+\mathbf{b}+\mathbf{c}|=1$ and $\mathbf{a}$ is perpendicular to $\mathbf{b}$. If $\mathbf{c}$ makes angles $\alpha, \beta$ with $\mathbf{a}, \mathbf{b}$ respectively, then $\cos \alpha+\cos \beta=$

If $\mathbf{a}$ is a vector such that $\mathbf{a} \times \hat{\mathbf{i}}=\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{a} \cdot \hat{\mathbf{i}}=1$, then equation of the line passing through the point $\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and parallel to $\mathbf{a}$ is