For vectors $\mathbf{u}=u_x\mathbf{i}+u_y\mathbf{j}$ and $\mathbf{v}=v_x\mathbf{i}+v_y\mathbf{j}$: the dot product is a scalar; the cross product (in 2D) gives a vector perpendicular to the plane with magnitude $u_x{v}_y-u_y{v}_x$. The angle $\theta$ satisfies $\mathbf{u}\cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}|\cos \theta$.

Let $\mathbf{u}=3\mathbf{i}+4\mathbf{j},\mathbf{v}=2\mathbf{i}-5\mathbf{j}$.

Find:

1. Dot product $\mathbf{u}\cdot \mathbf{v}$
2. Magnitudes $|\mathbf{u}|$ and $|\mathbf{v}|$
3. Angle $\theta$ between $\mathbf{u}$ and $\mathbf{v}$
4. Cross product $\mathbf{u}\times \mathbf{v}$ (as a scalar multiple of $\mathbf{k}$)