Line integral with cross products

Question

I'm in AP Physics C, and some of the topics in the electricity and magnetism portion use multivariable or vector calculus. Since it's a physics class, the math always simplifies to simple calculus or geometry and you don't actually need to do vector calculus. However, I was curious about the math so I started teaching myself some multivariable calculus, and found out that, for instance, Gauss's Law: $$\oint\vec{E}\cdot\vec{dA} = \frac{Q_{enc}}{\varepsilon_{0}}$$ really uses a surface integral and that the formula for calculating the potential difference along a curve C $$\Delta V = - \int_{C}\vec{E}\cdot\vec{dr}$$ involves a line integral. Both of these integrals eventually evaluate to a scalar. However, when we went into magnetism, I ran into two equations that seemingly involved line integrals, but involved the integrand being crossed with a differential element, resulting in a vector answer. These equations are the Biot-Savart Law for calculating the magnetic field at a point P due to a wire that lies along a curve C $$\vec{B_{p}} = \frac{\mu_{0}I}{4\pi}\int_{C}\frac{\vec{ds}\times\hat{r}}{\left| \vec{r}\right|^{2}}$$ and the equation for the magnetic force on a wire that lies along a curve C due to an external magnetic field $$\vec{F_{B}} = \int_{C}I \vec{dl} \times \vec{B}$$ So far, all of the line integrals I have seen involved the integrand (a vector field or multivariable function) dotted with a differential curve element, so are these integrals even line integrals in the first place? Could you link me to a resource that shows what they are called and defines a mathematically rigorous way to evaluate them? The only integrals I've seen so far that produce vector answers are integrals of space curves as the parameter varies from a to b, and the textbook I was using didn't do a very good job of explaining what their significance is.