<s>[INST] How do we parametrize the curve determined by the intersection of $x^{2} + y^{2} + z^{2} = 1$ and $x + y + z = 1$ and find it arc length? How do we parametrize the curve determined by the intersection of $x^{2} + y^{2} + z^{2} = 1$ and $x + y + z = 1$?
Should I make a new equation $x^{2} + y^{2} + z^{2} = x + y + z$?
Or do I need to use spherical coordinates to parametrize $x^{2} + y^{2} + z^{2} = 1$ ?
Also, I want to find the arc length of this parametric equation.
 [/INST] The intersection will be an ellipse in 3-space.  Therefore, it is parameterisable by a single variable $t$.
First, solve the equation of the plane for $z$.
$$z = 1-x-y.$$
Then substitute into the equation of the sphere:
$$x^2+y^2+(1-x-y)^2 = 1.$$
Write this equation in the form:
$$\frac{(x-x_0)^2}{a^2}+\frac{(y-y_0)^2}{b^2} = 1.$$
Then, the parameterisation is given by
$$x = x_0 + a\cos(t),\quad y = y_0 + b\sin(t),\quad z = 1-x_0-y_0-a\cos(t)-b\sin(t), \quad 0 \leq t \leq 2\pi$$
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