Describe the locus of points in R3 equidistant from the origin (0, 0, 0), and
the plane x + y + z = 1.
A parabola is the locus of points equidistant from a point and a line.
If you rotate it around its axis of symmetry, you get a surface with points equidistant from a point and a plane: a paraboloid.
To describe the locus of points in R3 that are equidistant from the origin and the plane x + y + z = 1, we can use the concept of perpendicular distance.
The distance between a point (x, y, z) and the origin (0, 0, 0) can be calculated using the distance formula:
d = sqrt(x^2 + y^2 + z^2)
Similarly, the distance between a point (x, y, z) and the plane x + y + z = 1 can be calculated using the formula for the distance between a point and a plane:
d' = |ax + by + cz + d| / sqrt(a^2 + b^2 + c^2),
where a, b, c are the coefficients of the plane equation, and d is a constant term.
In our case, the equation of the plane is x + y + z = 1, so a = b = c = 1, and d = -1.
Now, to find the locus of points that are equidistant from the origin and the plane x + y + z = 1, we equate the two distances:
sqrt(x^2 + y^2 + z^2) = |x + y + z - 1| / sqrt(3)
Squaring both sides of the equation, we get:
x^2 + y^2 + z^2 = (x + y + z - 1)^2 / 3
Expanding the right-hand side, we have:
3(x^2 + y^2 + z^2) = (x + y + z - 1)^2
Expanding further, we get:
3(x^2 + y^2 + z^2) = x^2 + y^2 + z^2 + 1 + 2xy + 2xz + 2yz - 2(x + y + z)
Simplifying the equation, we have:
2(x^2 - xy - xz + y^2 - yz + z^2 - x - y - z) - 1 = 0
This equation represents the locus of points in R3 that are equidistant from the origin and the plane x + y + z = 1.