Introduction to Three Dimensional Geometry

🎯 Key Points

• Three mutually perpendicular axes meet at the origin; the three coordinate planes (XY, YZ, ZX) cut space into 8 octants
• A point is written P(x, y, z) - x is measured along OX, y along OY, z along OZ
• A point on the XY-plane has z = 0; on the YZ-plane x = 0; on the ZX-plane y = 0. On the x-axis both y = 0 and z = 0
• Distance PQ = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²] - the 2D formula with one extra term
• Distance of P(x, y, z) from the origin = √(x² + y² + z²)
• Internal section in ratio m:n → ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n), (mz₂+nz₁)/(m+n)); for external division replace n by −n
• Midpoint is the m:n = 1:1 case → ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2)
• Centroid of a triangle → ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3, (z₁+z₂+z₃)/3)

Coordinate Axes and Coordinate Planes

Take three mutually perpendicular lines through a fixed point O. These are the x-axis, y-axis and z-axis, and O is the origin. Taken in pairs they determine three planes:

• XY-plane - contains the x and y axes; every point on it has z = 0
• YZ-plane - contains the y and z axes; every point on it has x = 0
• ZX-plane - contains the z and x axes; every point on it has y = 0

These three planes divide space into eight regions called octants, distinguished by the sign pattern of (x, y, z). The first octant is where all three are positive.

Coordinates of a Point in Space

To locate P, drop perpendiculars from P onto the three axes. The signed lengths cut off are its coordinates (x, y, z). Some useful special cases:

• Origin → (0, 0, 0)
• Point on the x-axis → (x, 0, 0); on the y-axis → (0, y, 0); on the z-axis → (0, 0, z)
• Point in the XY-plane → (x, y, 0)

Distance Between Two Points

For P(x₁, y₁, z₁) and Q(x₂, y₂, z₂):

PQ = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]

This follows from applying Pythagoras twice - once in the base plane, once vertically. Setting the second point to the origin gives OP = √(x² + y² + z²).

Section Formula

If R divides the join of P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) internally in the ratio m : n, then

R = ( (mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n), (mz₂+nz₁)/(m+n) )

For external division in ratio m : n, replace n with −n throughout, giving denominators of (m − n). The midpoint is simply the case m = n = 1.

💡 Advanced Edge

• To test whether three points are collinear in 3D, compute all three pairwise distances - the largest must equal the sum of the other two. (Slopes don't exist in 3D, so the 2D slope test doesn't transfer.)
• A ratio that comes out negative when you solve the section formula means the point divides the segment externally, not internally.
• To find where the line joining two points crosses a coordinate plane, set the relevant coordinate to 0 in the section formula and solve for the ratio - e.g. for the XY-plane set the z-coordinate to 0, giving k = −z₁/z₂.

Example 1 - Distance
Q: Find the distance between P(1, −3, 4) and Q(−4, 1, 2).
Step 1 - Differences: Δx = −4 − 1 = −5, Δy = 1 − (−3) = 4, Δz = 2 − 4 = −2.
Step 2 - Square and add: 25 + 16 + 4 = 45.
Answer: PQ = √45 = 3√5 units.

Example 2 - Section formula
Q: Find the point dividing the join of A(1, −2, 3) and B(3, 4, −5) internally in the ratio 1 : 3.
Step 1 - Here m = 1, n = 3, so m + n = 4.
Step 2 - x: (1·3 + 3·1)/4 = 6/4 = 3/2.
Step 3 - y: (1·4 + 3·(−2))/4 = (4 − 6)/4 = −1/2.
Step 4 - z: (1·(−5) + 3·3)/4 = (−5 + 9)/4 = 1.
Answer: (3/2, −1/2, 1).

Example 3 - Crossing a coordinate plane
Q: In what ratio does the YZ-plane divide the line joining A(−2, 4, 7) and B(3, −5, 8)?
Step 1 - On the YZ-plane the x-coordinate is 0.
Step 2 - Let the ratio be k : 1, so x = (3k − 2)/(k + 1) = 0.
Step 3 - Solve: 3k = 2, so k = 2/3.
Answer: the YZ-plane divides AB internally in the ratio 2 : 3.

The Eight Octants

• The three mutually perpendicular coordinate planes (XY, YZ, ZX) divide space into eight regions called octants.
• Each octant is identified by the sign pattern of the coordinates (x, y, z); the first octant has all three positive.
• A point is located by its signed perpendicular distances from the three coordinate planes, giving its (x, y, z) coordinates.
• The point where the three axes meet is the origin (0, 0, 0), common to all octants.
• Knowing the octant of a point from its signs helps visualise its position without plotting.

Special Points on Axes and Coordinate Planes

• A point on the x-axis has form (x, 0, 0); on the y-axis (0, y, 0); on the z-axis (0, 0, z) — the two coordinates off that axis are zero.
• A point in the XY-plane has z = 0, in the YZ-plane has x = 0, and in the ZX-plane has y = 0.
• These conditions are handy for finding where a line or figure meets an axis or a coordinate plane.
• The foot of the perpendicular from a point (x, y, z) to the XY-plane is (x, y, 0), obtained by dropping the z-coordinate to zero.
• Distances from a point to the coordinate planes are simply the absolute values of the corresponding coordinates.

Midpoint and Centroid Formulae in Space

• The midpoint of the segment joining (x₁, y₁, z₁) and (x₂, y₂, z₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2).
• The midpoint is the special case of the section formula for the ratio 1 : 1.
• The centroid of a triangle with vertices (x₁, y₁, z₁), (x₂, y₂, z₂), (x₃, y₃, z₃) is ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3, (z₁ + z₂ + z₃)/3).
• The centroid divides each median of the triangle in the ratio 2 : 1 from the vertex.
• These averaging formulae extend the familiar two-dimensional results by simply adding the z-coordinate.

Collinearity of Three Points in Space

• Three points A, B, C in space are collinear if they lie on a single straight line.
• A direct test uses distances: the points are collinear when the largest of AB, BC, CA equals the sum of the other two.
• Equivalently, C divides AB in some ratio, so the section formula must reproduce the coordinates of C for a suitable ratio k : 1.
• If no such ratio exists, or if the three pairwise distances cannot satisfy the sum condition, the points form a genuine triangle.
• Collinearity checks are common when verifying that a point lies on the line through two given points.