geometry

Three-dimensional coordinate geometry: Formulas ,Examples

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1 month ago

January 31, 2025

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Three-dimensional coordinate geometry

Three-dimensional coordinate geometry is the mathematical study of points, lines, planes, and forms in three dimensions. Unlike two-dimensional (2D) geometry, which deals with figures on a flat plane, 3D geometry expands into space, adding a third coordinate (depth or height).

Key Concepts of Three-dimensional Geometry

1. Coordinate system in 3D

A point in space is represented as (x, y, z) in cartesian coordinate system , Where

x is the distance along the X axis (horizontal)
y is the distance along the Y axis (vertical)
z is the distance along the Z axis (depth)

2. Distance between two points

Given two points, A( x1,y1,z1) and B(x2,y2,z2), the distance between them is:

3. Section formula

If a point P(x,y,z) splits the line segment connecting 𝐴(𝑥1, 𝑦1, 𝑧1) and 𝐵(𝑥2, 𝑦2, 𝑧2) in the ratio m:n, then:

4. Equation of a line in 3D

Vector form

In this equation, r_0 represents a point’s position on the line, 𝑑 represents the direction vector, and 𝜆 represents a scalar.

Cartesian form

If a line passes through (𝑥1, 𝑦1, 𝑧1) and has direction ratios a,b,c, its equation is

5. Equation of a plane in 3D

General form:

where A, B, and C are the normal direction ratios.

Vector form:

In this equation, n is the normal vector, r is a point’s position vector on the plane, and d is a constant.

Plane through three points

If a plane passes through $A (x_{1}, y_{1}, z_{1})$ , $B (x_{2}, y_{2}, z_{2})$ , and C(x3,y3,z3), its equation is found using the determinant

6. Angle between two lines or planes

Between two lines: If two lines have direction vectors d1=(a1,b1,c1) and d2=(a2,b2,c2), the angle between them is

Between two planes: If two planes have direction vectors n1 = (A1, B1, C1) and n2 = (A2, B2, C2), the angle between them is

7. Intersection of a line and a plane

To find the intersection point, substitute the line’s parametric equations into the plane equation and solve for λ.

8. Shortest distance between two skew lines

Two skew (non-parallel, non-intersecting) lines can be written as

The shortest distance formula is given by

where r1, r2are position vectors and d1, d2re direction vectors.

9. Direction consines and Direction ratios in 3D geometry

Direction Ratios: Any three proportional values, a, b, and c, that specify the line’s direction are its direction ratios. The direction ratios of a line that goes through two points A(x1,y1,z1) and B(x2,y2,z2) are as follows if it passes through both places:

These values represent the difference in coordinates between two points in the line.

Direction Cosines: The cosines of the angles a line makes with the coordinate axes are its direction cosines. The direction cosines are as follows if a line forms angles α,β, and γ with the X, Y, and Z axes, respectively:

These cosines satisfy the fundamental equation :

Direction Ratios and Direction Cosines’ Relationship

If a line has direction ratios a, b, c, then its direction cosines are

Where is the magnitude of the direction vector.

Three-dimensional coordinate geometry Examples

Three-dimensional Example 1

Find the direction cosines and direction ratios of the straight lines OP,OQ,PQ where P(2,-3,4),Q(-1,2,3),O(0,0,0)

Solution

Use these procedures to determine the direction cosines and direction ratios of the straight lines OP, OQ, and PQ.

The direction ratios of a line joining two points (x1, y1, z1) and (x2, y2, z2) are given by

For OP (O(0,0,0) to P(2,-3,4)): Direction ratios = (2−0,−3−0,4−0)=(2,−3,4)
For OQ((O (0,0,0) to Q(-1,2,3)): Direction ratios= (-1−0,2−0,3−0)=(-1,2,3)
For PQ (P(2,-3,4) to Q(-1,2,3)): Direction ratios= (-1-2, 2-(-3), 3-4) = (-3, 5,-1)

The directions cosines (l, m, n) are given by

Where a, b, and c are the direction ratios.

For Op (2,-3,4)

Magnitude =

For OQ (-1,2,3)

Magnitude =

For PQ (-3,5,-1)

Magnitude =

Example 2

If the direction cosines of two lines are connected by the relation 2l+2m-n=0 and lm+mn+nl=0, then find the direction cosines of the lines. Show that the lines are perpendicular to each other.

Solution

We are given that the direction cosines (l,m,n) of two lines satisfy the equations:

2l+2m-n=0

lm+mn+nl=0

Step 1: Explain One Variable Using Other Variables

From the first equation

n=2l+2m

Step 2: In the Second Equation, substitute

Adding n = 2l+2m to the second equation:

lm+m(2l+2m)+l(2l+2m)=0