Three‑Dimensional Geometry

Vector Representation of a Point  

A point in space can be represented by a position vector, which shows its location relative to the origin along the x, y, and z axes.  

Distance in 3D  

The distance between two points is measured by considering differences along all three axes. Similarly, the distance of a point from each coordinate axis is its perpendicular distance to that axis.  

Section Formula  

This explains how a point divides a line segment in a given ratio. The midpoint is a special case where the ratio is equal.  

Direction Cosines and Ratios  

- Direction cosines are the cosines of the angles a line makes with the x, y, and z axes.  

- Direction ratios are numbers proportional to these cosines.  

- They help in defining the orientation of a line in space.  

Angle Between Two Lines  

The angle between two lines depends on their orientation. Lines can be parallel, perpendicular, or intersecting depending on their direction ratios.  

Projection of a Line Segment  

Projection shows how much of one line segment lies in the direction of another line.  

Equation of a Plane  

A plane in 3D can be represented in different forms:  

- General form (using coefficients of x, y, z).  

- Normal form (based on perpendicular distance from origin).  

- Point form (plane passing through a given point).  

- Intercept form (plane cutting the axes at certain points).  

- Vector form (using position and normal vectors).  

Plane and Point Relations  

- The perpendicular distance of a point from a plane can be defined.  

- The foot of the perpendicular is the closest point on the plane.  

- The image of a point with respect to a plane is its reflection across the plane.  

Angle Between Two Planes  

The angle depends on the orientation of their normal vectors. Planes can be parallel or perpendicular depending on these normals.  

Angle Bisectors Between Planes  

The bisector planes divide the angle between two given planes into equal parts.  

Family of Planes  

A family of planes refers to all planes passing through the intersection of two given planes.  

Area and Volume  

- The area of a triangle in 3D can be determined using vectors.  

- The volume of a tetrahedron is calculated using the coordinates of its vertices.  

Equation of a Line  

A line in 3D can be represented as:  

- Intersection of two planes.  

- Symmetric form (ratios along axes).  

- Vector form (position vector plus scalar multiple of direction vector).  

Angle Between Line and Plane  

The angle is determined by the relationship between the line’s direction and the plane’s normal.  

Condition for a Line to Lie in a Plane  

A line lies in a plane if its point satisfies the plane equation and its direction is consistent with the plane’s orientation.  

Skew Lines  

Skew lines are non‑parallel, non‑coplanar lines that never intersect. Their shortest distance is defined using vector methods.