Introduction to Three-Dimensional Geometry: Distance, Important Formulae

Three- dimensional geometry is important to understand the different types of shapes and figures.

• In the real world, almost everything we come across is in three dimensions.
• Even a thin piece of paper or a strand of hair has some thickness, if you look at them sideways.
• Even in mathematics, three-dimensional Geometry (3D Geometry) plays an important role.
• It consists of 3 coordinates which are x-coordinate, y-coordinate, and z-coordinate.
• In a 3D space, three parameters are required to find the exact position of a point.

Also Read: NCERT Solutions For Class 12 Mathematics Chapter 10 Vector Algebra 

Key Terms: Three-dimensional plane, Vectors, Magnitude, Dimensions, One-dimensional geometry, Angle equation, Distance, 3D space

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Three- Dimensional Geometry

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In mathematics, Geometry is one of the most important topics.

• The concepts of Geometry are defined with respect to the planes.
• Geometry is divided into three categories based on their dimensions which are one-dimensional geometry, two-dimensional geometry, and three-dimensional geometry.
• A coordinate system is a method of determining the position or location of a point on the coordinate plane in three dimensions.
• Coordinate geometry contains all of the fundamental concepts, theorems, and formulae related to coordinate or analytic geometry.
• In this Chapter, we will learn about the cosine ratios of two lines as well as planes, the angle between two lines and also for planes, the shortest distance between two skew lines, and the distance from the point, etc.
• To simplify the solution to these problems, the results obtained in the vector form are converted to the cartesian equation.

The video below explains this:

Three-Dimensional Geometry Lines Detailed Video Explanation:

Read More: Addition of Vectors

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Direction Cosines and Direction Ratios of Line

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Assume that a line L is passing through the three-dimensional plane.

• Now, x,y, and z are the axes of the plane and α,β, and γ are the three angles the line makes with these axes.
• These are called the direction angles of the Plane.
• So, accordingly, we can say that cosα, cosβ, and cosγ are the direction cosines of the given line L.

Now If we give different names to the direction cosines and direction ratios to make the calculations simpler like, a,b, and c are the direction ratios and x,y, and z are the direction cosines then we can derive the formula as below.

x/a + y/b + z/c = k

Where k is any constant

So, 

x=ak, y=bk and z=ck

But as we know that x²+y²+z²=1

 So, as per the above equation 

(ak)²+(bk)²+(ck)²=1

k²=1/(a²+b²+c²)

So,

These are the direction cosines of the given line. Where on the basis of the desired sign of k, either a positive or negative sign is to be taken for x, y, and z.

Read More: Equation of a Plane in Three-Dimensional Space

Example. A Given line L has direction ratios -18, 12, and -4 then find out the direction cosines of the given line L.

Ans. As we know in the above-derived equation x, y and z are the direction cosines. But Here in this example a=-18, b=12 and c=-4

So, 

Now for y, 

Similarly for z,

z= c/(a²+b²+c²)

z= -4/(324+144+16)

z=-4/22

z=-2/11

So, direction cosines are -9/11,6/11 and -2/11.

Read More: Coplanar Vector

Equation of Line Passing through only one Point

If x,y, and z are the direction cosines of the line then the equation of the line is-

Where x,y, and z are the coordinates of the given point and x₁,y₁ and z₁ are the coordinates of any point on that line.

Read More: Edges, Faces, and Vertices

Equation of Line Passing Through Two Points

The position vector of two points A (x₁,y₁, z₁) and B (x₂,y₂, z₂) respectively are lying on a line. So, the equation of the line passing through these two points can be given as-

Where x,y, and z are the coordinates of a given point and (x₁,y_1, and z₁) and (x₂,y_2, and z₂) are the coordinates of points on the Line.

Read More: Coordinate Geometry

Angle Between Two Lines

Consider there are two lines named L1 and L2 which are passing through the three-dimensional plane. In which x₁,y_1, and z₁ and x₂, y_2, and z₂ are the direction ratios of two lines L₁ and L₂ respectively.

If we consider an angle between two lines is then

So, now if we have to find the sin then

On deriving this equation we get,

Concept of Plane and Equation of Plane in Normal Form

A plane is determined when the distance of the plane from the origin is given and it also passes through the given point and is also perpendicular to the given directions. The plane also passes through three non Collinear points.

The Equation of the normal form is defined as,

ax+by+cz=d

Where a,b and c are the direction cosines of the given vector.

And x,y and z are the coordinates of a given point on the plane and d is a distance from the origin.

Also Read:

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Things To Remember

• If two planes are perpendicular to each other then the product of the two vectors is always Zero.
• If two planes are at a right angle then the angle between them is always 90 degrees.
• We can also say that the direction ratios of two lines are the numbers that are proportional to the direction cosines of the line.
• We can define a skew line as a line in the space that is neither parallel nor intersecting.
• If we have to find the shortest distance between two skew lines then the segment is perpendicular to both the lines.
• The Equation of the normal form is defined as an ax+by+cz=d
• The vector equation of the line is always passing through the given point whose position is always fixed.
• We can say that two lines are coplanar if and only if the answer of the equation of both the lines is zero.

Read More: Coplanarity