3d Geometry

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Introduction to 3 Dimensional Geometry

3 Dimensional geometry involves the mathematics of shapes in 3D space and involves 3 coordinates in the XYZ plane which are x-coordinate, y-coordinate and z-coordinate. The shapes that occupy space are called 3D shapes. 3D shapes can also be defined as the solid shapes having three dimensions length, width, and height. Three-dimensional space is a geometric three-parameter model in which (there are three axes x,y, and z-axes) all known matter exists. These three dimensions are chosen from the terms length, width, height, depth, and breadth.

In a 3d space, three parameters are required to find the exact location of a point. 3-dimensional geometry plays a major role in JEE exams as a lot of questions are included in the exam. In this article, we will study the basic concepts of geometry involving 3-dimensional coordinate geometry which will help to understand different operations on a point in 3d plane.

Coordinate System in 3D Geometry

In 3 dimensional coordinate geometry, a coordinate system refers to the process of identifying the position or location of a point in the coordinate plane. To understand more about coordinate planes and systems, refer to the coordinate geometry lesson which covers all the basic concepts, theorems, and formulas related to coordinate or analytic geometry.

The Properties of Three-dimensional Space:

  • A point is represented by an ordered triple (x, y, and z) that consists of three numbers, an x-coordinate, a y-coordinate, a z-coordinate.

  • In the two-dimensional xy-plane, these coordinates indicate the signed distance along the coordinate axes,

  • The x-axis, y-axis, and z-axis, respectively, originate from the origin, denoted by O, which has coordinates (0, 0, and 0). 

Rectangular Coordinate System in Space

The coordinate system defines the position of a vector. In the rectangular coordinate in space, we refer to the three-dimensional space. To demonstrate the position of a vector mark a point as the origin, represented by the point ‘O’. The distance of any vector is now measured from this standard point.

Let ‘O’ be any point in space called origin and X’OX, Y’OY and Z’OZ be three lines perpendicular to each other and these three lines denote the coordinate  X, Y, and Z-axis. The planes XY, YZ, and ZX are called the coordinate planes in space.   

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Coordinates of a Point in Space

Consider a point P in space. The position of the point P is given by the coordinates (x, y, z) where x, y, z denote the perpendicular distance from YZ-plane, ZX-plane, and XY-plane respectively. If the vectors i, j, k  are assumed to be the unit vectors along OX, OY, OZ respectively, then the position vector of point P is xi + yj + zk or simply (x, y, z).

If ‘O’ is the origin and P is any point with coordinates (x, y, z) from the origin then the distance vector OP by the distance formula is given by OP = √x2 + y2 + z2.

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Some Key Points to Remember

  • The formula for distance between the points P(x1, y1, z1) and Q (x2, y2, z2) is √(x1 – x2)2 + (y1 – y2)2 + (z1 – z2)2

  • The point dividing the line joining the points P(x1, y1, z1) and Q (x2, y2, z2) in m : n ratio is 

(mx2 – nx1)/(m + n), 

(my2 – ny1)/(m + n),

(mz2 – nz1)/(m + n) where m + n ≠ 0. 

  • The coordinates of the centroid of a triangle having vertices A (x1, y1, z1), B (x2, y2, z2) and C (x3, y3, z3) is G ((x1 + x2 + x3) / 3, (y1 + y2 + y3) / 3, (z1 + z2 + z3)/ 3). 

Direction Cosines of a Line

The cosines of the angles made by a directed line segment with the coordinate axes are called the direction cosines of that line.

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As shown in the figure above, if α, β, and γ are the angles made by the line segment with the coordinate axes then these angles are said as direction angles; and the cosines of these directional angles are the direction cosines of the line. 

Also, cos α, cos β, and cos γ are called the direction cosines and are denoted by l, m, and n respectively.

l = cos α, 

m = cos β and 

n = cos γ

If three numbers are in proportion with the direction cosines of a line then they are called the direction ratios. Hence, if ‘a’, ‘b’, and ‘c’ are the direction ratios, and l, m, n are the direction cosines then, we must have,

a/l = b/m = c/n.

Some Key Points of Direction Cosines

  • As we know that l = cos α, m = cos β, and n = cos γ also -1< cos x< 1 ∀ x ∈ R, so l, m, and n are real numbers with values varying between -1 to 1. So, direction cosine’s ∈ [-1,1].

  • The angles between the x-axis and the coordinate axes are 0°, 90°, and 90°. So the direction cosines are cos 0°, cos 90° and cos 90° i.e. 1, 0, 0 respectively.

  • The direction cosine of the x, y, and z axes are (1,0,0), (0,1,0) and (0,0,1).

  • The direction cosines of a line parallel to any coordinate axis are equal to the direction cosines of the corresponding axis.

  • The direction cosines are associated by the relation l2 + m2 + n2 =1.

  • If the given line is inverted, then the dc’s will be cos (π − α), cos (π − β), cos (π − γ) or − cos α, − cos β, − cos γ.

  • Thus, a line can have two sets of direction cosines according to its direction.

  • If the two lines are parallel then their direction cosines are always the same.  

  • Direction ratios and direction cosines are proportional to each other and hence for a given line, there can be infinitely many direction ratios.

Perpendicular Distance of a Point from a Line

Let us assume AB be the straight line passing through the point A (a, b, c) and having direction cosines l, m, and n. Now, if AN is the projection of line AP on the straight line AB then we have,

AN = l(x – a) + m(y – b) + n(z – c),

and AP = √(x–a)2 + (y–b)2 + (z–c)2 

∴ Perpendicular distance of point P

PN = √(AP2 – AN2)

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The length of the perpendicular from a point (x1, y1, z1) to a plane ax+ by + cz + d = 0 is p = [(ax1 + by1 + cz1 + d)/ √(a2 + b2 + c2) ]