In Focus: 3D Geometry

In Focus : 3D Geometry

The Coordinate Geometry is a vast subject in the field of Mathematics. The expanse of coordinate geometry is infinite as more and more dimensions are being discovered in the real world. However, we now limit our studies upto only three dimensions. This article will deal with all the basics to charge you up for 3D Geometry!

The topic of 3D geometry is quite important and a bit complicated as compared to its counterpart two dimensional geometry. One of the ways of describing a 3D object is by approximating or assuming its shape as a mesh of triangles. A triangle is generally defined by three vertices wherein the positions of the vertices are described by the coordinates x, y and z. The major heads that are included in 3D coordinate geometry are the direction ratios and direction cosines of a line segment along with definitions of the plane.

Direction Cosines of a Line Segment

The direction cosines are the cosines of the angles between a line and the coordinate axis. If we have a vector (a, b, c) in three dimensional space, then the direction cosines of the vector are defined as

cos α = a/ √(a² + b² + c²), cos β = b/ √(a² + b² + c²), cos γ = c/ √(a² + b² + c²)

Direction Ratios

If l, m and n are the direction cosines then the direction ratios say a, b and c are given by

Points to remember:

• While the direction cosines of a line segment are always unique, the direction ratios are never unique and in fact they can be infinite in number.
• If the direction cosines of a line are l, m and n then they satisfy the relation l² + m² + n² = 1.

• If the direction cosines of a line segment AB are (l, m, n) then those of line BA will be (-l, -m, -n).

Angle Between Two Lines

Let us assume that θ is the angle between the two lines say AB and AC whose direction cosines are l₁, m₁ and n₁ and l₂, m₂ and n₂ then

cos θ = l₁l₂ + m₁m₂ + n₁n₂

Also if the direction ratios of two lines a₁, b₁ and c₁ and a₂, b₂ and c₂ then the angle between two lines is given by

cos θ = (a₁a₂ + b₁b₂ + c₁c₂)/ √(  a₁² + b₁² + c₁²) . √ (a₂² + b₂² + c₂²)

Then what is the Condition for Parallel or Perpendicular Lines?

When the two lines are perpendicular, the angle between the lines is 90° which gives the condition of perpendicularity as

l₁l₂ + m₁m₂ + n₁n₂ = 0

or this implies a₁a₂ + b₁b₂ + c₁c₂ = 0.

Similarly, when the two lines are parallel, the angle between them i.e. θ = 0.

This gives l₁/l₂ = m₁/m₂ = n₁/n₂

This also gives a₁/a₂ = b₁/b₂ = c₁/c₂

Projection of a line segment on a given line

Suppose we have a line segment joining the points P (x₁, y₁, z₁) and Q(x₂, y₂, z₂), then the projection of this line on another line having direction cosines as l, m, n is AB =  l(x₂-x₁) + m(y₂-y₁) + n(z₂-z₁).

Plane

A plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point(zero dimensions), a line(one dimension) and three-dimensional space. A plane in three-dimensional space has the equation

ax + by + cz + d = 0

where at least one of the numbers and must be non-zero. A plane in the 3D coordinate space is determined by a point and a vector that is perpendicular to the plane.

A plane in the 3D coordinate space is determined by a point and a vector that is perpendicular to the plane. Let P₀ = (x_o,y_o,z_o) be the point given, and n the orthogonal vector. Also, let P = (x,y,z) be any point in the 3D space, and r and ro the position vectors of the points P and P_o respectively. Now, if we let n = (a,b,c)  then since P_oP is perpendicular to n we have