Three-dimensional coordinates let us describe locations in space using ordered triples like (x, y, z). This system is essential for geometry, physics, engineering, computer graphics, and navigation because real objects have length, width, and height. The x-, y-, and z-axes meet at the origin O(0, 0, 0), giving every point in space a unique address.
A 3D coordinate grid helps turn spatial relationships into numbers that can be measured and calculated.
To plot a point (x, y, z), move x units along the x-axis, y units parallel to the y-axis, and z units parallel to the z-axis. Dashed guide lines to the coordinate planes show how the point relates to its coordinates. The distance between two 3D points comes from applying the Pythagorean theorem in two directions, then the third.
The signs of x, y, and z also place a point into one of eight regions called octants.
Key Facts
- A point in 3D space is written as an ordered triple (x, y, z).
- The origin is O(0, 0, 0), where the x-, y-, and z-axes intersect.
- The coordinate planes are the xy-plane where z = 0, the xz-plane where y = 0, and the yz-plane where x = 0.
- Distance from the origin to (x, y, z) is d = sqrt(x^2 + y^2 + z^2).
- Distance between P1(x1, y1, z1) and P2(x2, y2, z2) is d = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2).
- The eight octants are determined by the positive or negative signs of x, y, and z.
Vocabulary
- Ordered triple
- An ordered triple is a set of three numbers (x, y, z) that gives the position of a point in three-dimensional space.
- Origin
- The origin is the point O(0, 0, 0) where all three coordinate axes meet.
- Coordinate axis
- A coordinate axis is one of the three perpendicular number lines used to measure position in 3D space.
- Coordinate plane
- A coordinate plane is a flat plane formed by two coordinate axes, such as the xy-plane, xz-plane, or yz-plane.
- Octant
- An octant is one of the eight regions into which the three coordinate planes divide 3D space.
Common Mistakes to Avoid
- Swapping the order of coordinates: The point (2, 5, 1) is not the same as (5, 2, 1) because x, y, and z give movements in different directions.
- Forgetting the z-coordinate: A point in three dimensions needs three numbers, so writing only (x, y) places the point in a plane, not in space.
- Using the 2D distance formula for 3D points: The z-difference must be included, so use d = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2).
- Confusing coordinate planes with axes: The xy-plane is a flat surface where z = 0, while the x-axis is a line where y = 0 and z = 0.
Practice Questions
- 1 Plot the point A(3, -2, 4) on a 3D coordinate grid. State which coordinate planes its guide lines would touch and give the perpendicular distance from A to each coordinate plane.
- 2 Find the distance between P(1, 2, -3) and Q(5, -1, 3).
- 3 A point has coordinates (-4, 0, 6). Explain whether it lies in an octant, on an axis, or in a coordinate plane, and justify your answer.