Vectors Questions

The resultant of two forces $P\text{N}$ and $3\text{N}$ is a force of $7\text{N}$. If the direction of $3\text{N}$ force were reversed, the resultant would be $\sqrt{19}\text{N}$. The value of $P$ is:

If a line makes an angle of $\pi /4$ with the positive directions of each of $x$ -axis and $y$ -axis, then the angle that the line makes with the positive direction of the $z$ -axis is:

If $\hat{u}$ and $\hat{v}$ are unit vectors and $\theta$ is the acute angle between them, then $2\hat{u}\times 3\hat{v}$ is a unit vector for:

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{b}=\hat{i}-\hat{j}+2\hat{k}$ and $\vec{c}=x\hat{i}+(x-2)\hat{j}-\hat{k}$. If the vector $\vec{c}$ lies in the plane of $\vec{a}$ and $\vec{b}$, then $x$ equals: