1.1 Three-Dimensional Coordinate Systems

Three-Dimensional Coordinate Systems

Three-dimensional coordinate systems expand our understanding of space beyond flat surfaces. They use three perpendicular axes to pinpoint locations, creating a framework for describing position, movement, and relationships in 3D. These tools show up constantly throughout multivariable calculus, so getting comfortable with them now pays off in every later unit.

Components of 3D Coordinate Systems

A 3D coordinate system is built from three mutually perpendicular axes that all intersect at the origin, the point $(0, 0, 0)$.

• The x-axis extends horizontally (left-right)
• The y-axis extends vertically (up-down)
• The z-axis extends into and out of the plane (depth)

Different textbooks orient these axes differently, so always check which convention your course uses. The standard convention follows the right-hand rule: point your right thumb along the positive x-axis, curl your fingers toward the positive y-axis, and your middle finger (or palm) points along the positive z-axis.

The three axes create three coordinate planes, each defined by setting one coordinate equal to zero:

• xy-plane $(z = 0)$: the "floor" where the x- and y-axes live
• xz-plane $(y = 0)$: a vertical plane running front-to-back
• yz-plane $(x = 0)$: a vertical plane running side-to-side

These three planes divide all of 3D space into eight regions called octants. The first octant is where all three coordinates are positive $(+x, +y, +z)$. The other seven octants correspond to the other sign combinations.

Points in 3D Space

Every point in 3D space is identified by an ordered triple $(x, y, z)$. Each coordinate measures a signed distance from one of the coordinate planes:

• The x-coordinate measures distance from the yz-plane
• The y-coordinate measures distance from the xz-plane
• The z-coordinate measures distance from the xy-plane

Positive values indicate one direction from the origin; negative values indicate the opposite direction.

A few special cases are worth recognizing:

• A point on a coordinate plane has one zero coordinate. For example, $(3, 0, 5)$ lies on the xz-plane because its y-coordinate is zero.
• A point on a coordinate axis has two zero coordinates. For example, $(0, 4, 0)$ lies on the y-axis.

Distance Between 3D Points

The distance formula in 3D is a direct extension of the 2D version, with an extra term for the z-coordinate:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

This comes from applying the Pythagorean theorem twice: once in a coordinate plane to get the diagonal across two dimensions, then again using that diagonal and the third dimension.

Example: Find the distance between $A = (1, -2, 3)$ and $B = (4, 0, -1)$.

1. Subtract corresponding coordinates: $(4-1) = 3$, $(0-(-2)) = 2$, $(-1-3) = -4$
2. Square the differences: $9, 4, 16$
3. Sum the squares: $9 + 4 + 16 = 29$
4. Take the square root: $d = \sqrt{29}$

Midpoints of 3D Line Segments

The midpoint of a segment connecting two points is found by averaging each pair of coordinates:

$M = \left(\frac{x_1 + x_2}{2},\; \frac{y_1 + y_2}{2},\; \frac{z_1 + z_2}{2}\right)$

This works for the same reason it does in 2D: the midpoint sits exactly halfway along each axis direction.

Example: The midpoint of $A = (1, -2, 3)$ and $B = (4, 0, -1)$ is:

1. Average x-values: $\frac{1+4}{2} = \frac{5}{2}$
2. Average y-values: $\frac{-2+0}{2} = -1$
3. Average z-values: $\frac{3+(-1)}{2} = 1$

So $M = \left(\frac{5}{2},\; -1,\; 1\right)$.

You can verify a midpoint by checking that it's equidistant from both endpoints using the distance formula.