Consider two n-dimensional vectors $\mathbf{x, x_0}$ and the expression $$ \int_0^\infty f(t) \delta(\mathbf x - t\mathbf x_0) dt $$ where $\delta$ is the n-dimensional Dirac delta function. Intuitively this means that $\mathbf x$ is collinear with $\mathbf x_0$ and that there is a weight $f(t)$ where $t$ is the "slope".
Now $\mathbf x = t \mathbf x_0, t > 0$ iff $\mathbf{x\cdot x_0} = \mathbf{|x||x_0|}$ with $t = \frac{\mathbf{x\cdot x_0}}{|\mathbf x_0|^2}$
So I am naively wondering if I may write $$ \int_0^\infty f(t) \delta(\mathbf x - t\mathbf x_0) dt = f\left( \frac{\mathbf{x\cdot x_0}}{|\mathbf x_0|^2} \right) \delta(\mathbf{x\cdot x_0 - |x||x_0|}) = f\left( \frac{|\mathbf x|}{|\mathbf x_0|} \right) \delta(\mathbf{x\cdot x_0 - |x||x_0|}) $$ and also whether there might even be a simpler expression?
Many thanks in advance! p