$A$ unit vector in the $xy$-plane which is perpendicular to $4i - 3j + k$ is

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{3}$. If $\lambda \overrightarrow{a} + 2 \overrightarrow{b}$ and $3 \overrightarrow{a} - \lambda \overrightarrow{b}$ are perpendicular to each other,then the number of values of $\lambda$ in $[-1, 3]$ is:

If $|a| = |b| = 1$ and $|a + b| = \sqrt{3}$,then the value of $(3a - 4b) \cdot (2a + 5b)$ is

If $G(\vec{g}), H(\vec{h})$ and $P(\vec{p})$ are the centroid,orthocenter,and circumcenter of a triangle respectively,and $x \vec{p} + y \vec{h} + z \vec{g} = 0$,then $(x, y, z) = $

Consider two vectors $\overrightarrow{u} = 3\hat{i} - \hat{j}$ and $\overrightarrow{v} = 2\hat{i} + \hat{j} - \lambda\hat{k}$,where $\lambda > 0$. The angle between them is given by $\cos^{-1}\left(\frac{\sqrt{5}}{2\sqrt{7}}\right)$. Let $\vec{v} = \vec{v}_1 + \vec{v}_2$,where $\vec{v}_1$ is parallel to $\overrightarrow{u}$ and $\vec{v}_2$ is perpendicular to $\overrightarrow{u}$. Then the value $|\vec{v}_1|^2 + |\vec{v}_2|^2$ is equal to

The shortest distance between the lines $r = 3i + 5j + 7k + \lambda(i + 2j + k)$ and $r = -i - j - k + \mu(7i - 6j + k)$ is