I am a high school student writing a maths exploration paper. For this paper I wanted to be able to find the point in space on a curved line, for instance a circle. Is it possible to obtain a vector equation for this or do alternate methods exist to model this?
I have done preliminary research on the internet, yet I was not able to find any other maths that could help me with this.
A typical curve $\gamma$ in ${\mathbb R}^3$ is described by a function $$\gamma:\quad[a,b]\to{\mathbb R}^3,\qquad t\mapsto{\bf r}(t)=\bigl(x(t),y(t),z(t)\bigr)\ .$$ such a thing is called a parametric representation of $\gamma$. If your $\gamma$ is a circle you have to know the center ${\bf c}=(c_1,c_2,c_3)$ and the radius $r>0$ of this circle, and above all: the unit normal ${\bf n}$ of the plane through ${\bf c}$ containing this circle. Given ${\bf n}$ it is a problem of linear algebra to construct two mutually orthogonal unit vectors ${\bf u}=(u_1,u_2,u_3)$ and ${\bf v}=(v_1,v_2,v_3)$ in this plane. When you have found one such vector ${\bf u}\perp{\bf n}$ you can put ${\bf v}:={\bf n}\times{\bf u}$. After ${\bf u}$ and ${\bf v}$ have been constructed a "vectorial" parametric representation of your circle $\gamma$ would be $$\gamma:\quad[0,2\pi]\to{\mathbb R}^3,\qquad t\mapsto {\bf c}+r\bigl(\cos t\> {\bf u}+\sin t\>{\bf v}\bigr)\ .$$ Of course you can write this out in the three coordinates separately.