Three Dimensional Geometry — Class 12 Maths Notes
Three Dimensional Geometry · Class 12 Maths · 5 topics.
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Topics covered in Three Dimensional Geometry
1.Introduction of Three Dimensional Geometry
Introduction to Three Dimensional Geometry
Three dimensional geometry, often referred to as 3D geometry, deals with shapes and objects that exist in three-dimensional space. Unlike two-dimensional shapes like squares and circles, which lie flat on a plane, three-dimensional shapes have depth as well as width and height. Common examples of 3D shapes are cubes, spheres, and pyramids.
Real-Life Examples
Imagine you're trying to pack for a trip and you need to choose the right suitcase. Understanding the 3D geometry of the suitcases (considering their length, width, and height) and how much they can hold (volume) helps you make the best choice. Or consider a cricket or football field; understanding the distances and the layout of the field in three dimensions can give players a competitive advantage.
Why It's Useful
- Daily Life Applications: From architecture and engineering to video games and virtual reality, 3D geometry helps in designing and interacting with objects in a space. It's used in planning rooms, buildings, and even entire cities.
- Career Implications: Careers in mechanical design, civil engineering, architecture, video game design, and many areas of science heavily rely on 3D geometry. For example, architects use 3D geometry to design buildings and ensure they are safe, functional, and aesthetically pleasing.
- Mathematical Understanding: It helps deepen your understanding of spatial relationships and geometry. This can improve your problem-solving skills in both academic and real-world scenarios.
Hands-On Activity
Try building a simple 3D model using materials like clay or building blocks. Start with a basic shape like a cube or a pyramid, and as you get more comfortable, try more complex structures. Notice how each shape occupies space and think about how you can calculate their volume and surface area.
Three-dimensional geometry is not just about complex calculations but also about understanding the world around us in a more complete way. It's a fascinating part of mathematics that has a direct impact on our daily lives and future careers!
2.Direction Cosines and Direction Ratios of a Line
In three-dimensional geometry, understanding the orientation of a line in space is crucial. This is where direction cosines and direction ratios come into play. They are tools that help us describe the angle and direction of a line with respect to the coordinate axes in 3D space.
Direction Cosines
Direction cosines (often abbreviated as "dcos") are the cosines of the angles that a directed line segment makes with the positive directions of the 𝑥x-, 𝑦y-, and 𝑧z-axes. These angles are usually denoted as 𝛼α, 𝛽β, and 𝛾γ. The direction cosines are given by:
- cos𝛼cosα (cosine of the angle with the 𝑥x-axis)
- cos𝛽cosβ (cosine of the angle with the 𝑦y-axis)
- cos𝛾cosγ (cosine of the angle with the 𝑧z-axis)
Mathematically, for a line with a direction vector 𝑎=(𝑎1,𝑎2,𝑎3)a=(a1,a2,a3), the direction cosines are calculated as: cos𝛼=𝑎1𝑎12+𝑎22+𝑎32,cosα=a12+a22+a32a1, cos𝛽=𝑎2𝑎12+𝑎22+𝑎32,cosβ=a12+a22+a32a2, cos𝛾=𝑎3𝑎12+𝑎22+𝑎32.cosγ=a12+a22+a32a3.
Direction Ratios
Direction ratios are simply the proportions of the components of any vector parallel to the line. They are denoted as 𝑎a, 𝑏b, and 𝑐c. They do not have a fixed magnitude but rather show the relative proportions in direction. For the direction vector 𝑎=(𝑎1,𝑎2,𝑎3)a=(a1,a2,a3), the direction ratios are 𝑎1a1, 𝑎2a2, and 𝑎3a3.
Real-Life Example
Consider a drone flying in the sky. To control its path, you need to know in which direction it's heading relative to the earth's surface. Using direction cosines, you can determine the exact angles the drone's path makes with the north (like the 𝑥x-axis), east (like the 𝑦y-axis), and vertically upwards (like the 𝑧z-axis). This helps in navigating the drone efficiently.
Hands-On Activity
Take a long stick or a pencil. Hold it in different positions like pointing towards different objects in your room (e.g., a lamp, a window, and the ceiling). Think of the pencil as a line in 3D space, and imagine the angles it makes with the floor and the walls. This simple activity helps visualize how direction cosines and ratios work in determining the orientation of a line in space.
Direction cosines and ratios are fundamental concepts in fields like aerospace, where understanding the orientation of an aircraft is crucial, and in computer graphics, where they help in rendering the orientation of objects in 3D models.
3.Equation of a Line in Space
Equation of a Line in Space
In three-dimensional geometry, the equation of a line in space is not as straightforward as in two-dimensional space. Unlike 2D, where a line is typically represented by a single linear equation, in 3D space a line is defined where two planes intersect, or alternatively, by parametric equations that involve direction cosines or direction ratios.
Vector and Parametric Forms of a Line
The vector form of a line in 3D is given by: 𝑟=𝑎+𝑡𝑏r=a+tb where:
- 𝑟r is the position vector of any point on the line,
- 𝑎a is the position vector of a fixed point on the line (often the point where the line intersects the origin plane),
- 𝑏b is a direction vector of the line (not necessarily a unit vector),
- 𝑡t is a scalar parameter.
The parametric form breaks down the vector equation into individual coordinates: 𝑥=𝑥0+𝑎𝑡,x=x0+at, 𝑦=𝑦0+𝑏𝑡,y=y0+bt, 𝑧=𝑧0+𝑐𝑡,z=z0+ct, where (𝑥0,𝑦0,𝑧0)(x0,y0,z0) are the coordinates of a known point on the line, and 𝑎,𝑏,𝑐a,b,c are the direction ratios of the line, corresponding to the direction vector.
Example Problems Based on the Equation of a Line in Space
Example 1: Find a point on the line Given the line equation 𝑥=1+2𝑡x=1+2t, 𝑦=−3+5𝑡y=−3+5t, 𝑧=4−𝑡z=4−t, find the coordinates of the point on this line when 𝑡=2t=2.
Solution: 𝑥=1+2(2)=1+4=5x=1+2(2)=1+4=5 𝑦=−3+5(2)=−3+10=7y=−3+5(2)=−3+10=7 𝑧=4−(2)=4−2=2z=4−(2)=4−2=2 So, the point is (5,7,2)(5,7,2).
Example 2: Determine if a point lies on a given line For the line equation 𝑥=3+4𝑡x=3+4t, 𝑦=1+2𝑡y=1+2t, 𝑧=2𝑡z=2t, does the point (11,5,2)(11,5,2) lie on this line?
Solution: Check each coordinate with the corresponding equation: 11=3+4𝑡 ⟹ 𝑡=211=3+4t⟹t=2 5=1+2𝑡 ⟹ 𝑡=25=1+2t⟹t=2 2=2𝑡 ⟹ 𝑡=12=2t⟹t=1 Since 𝑡t is not consistent across all three coordinates, the point does not lie on the line.
Example 3: Find the intersection of two lines Line 1: 𝑥=2+𝑡x=2+t, 𝑦=3+2𝑡y=3+2t, 𝑧=5𝑡z=5t Line 2: 𝑥=2+𝑠x=2+s, 𝑦=1+3𝑠y=1+3s, 𝑧=𝑠z=s Find if they intersect and the intersection point.
Solution: Set the equations equal to each other: 2+𝑡=2+𝑠2+t=2+s 3+2𝑡=1+3𝑠3+2t=1+3s 5𝑡=𝑠5t=s From 5𝑡=𝑠5t=s: 𝑠=5𝑡s=5t Substitute 𝑠=5𝑡s=5t into the first equation: 2+𝑡=2+5𝑡 ⟹ 4𝑡=0 ⟹ 𝑡=02+t=2+5t⟹4t=0⟹t=0 Substitute 𝑡=0t=0 into the second equation: 3=1+15×0 ⟹ 3≠13=1+15×0⟹3=1 Since there's no consistent solution, the lines do not intersect.
Real-World Application
Consider planning a new metro rail system where you need to establish the exact routes for tunnels that will not only travel horizontally across the city but also vertically between different depths underground. Using the equations of lines in 3D, engineers can determine the paths of these tunnels and ensure they do not intersect unintentionally with other underground structures.
4.Angle between Two Lines in 3D Geometry
Introduction to Angle between Two Lines in 3D Geometry
In three-dimensional geometry, the angle between two lines can be calculated using the concept of the dot product of their direction vectors. The angle is an essential aspect when determining how two objects, such as roads, pipes, or beams, intersect or align with each other in space.
Formula for Angle between Two Lines
The formula to find the angle 𝜃θ between two lines with direction vectors 𝑎=(𝑎1,𝑎2,𝑎3)a=(a1,a2,a3) and 𝑏=(𝑏1,𝑏2,𝑏3)b=(b1,b2,b3) is given by:
cos𝜃=𝑎⋅𝑏∣𝑎∣∣𝑏∣=𝑎1𝑏1+𝑎2𝑏2+𝑎3𝑏3𝑎12+𝑎22+𝑎32𝑏12+𝑏22+𝑏32cosθ=∣a∣∣b∣a⋅b=a12+a22+a32b12+b22+b32a1b1+a2b2+a3b3
This formula uses the dot product of vectors and the magnitudes of the vectors to calculate the cosine of the angle between them. This approach effectively captures the spatial relation between two lines in 3D.
Example 1: Calculate the Angle between Two Lines
Given Lines: Line 1 with direction vector 𝑎=(1,2,3)a=(1,2,3) Line 2 with direction vector 𝑏=(−1,0,4)b=(−1,0,4)
Solution: First, calculate the dot product of 𝑎a and 𝑏b: 𝑎⋅𝑏=(1)(−1)+(2)(0)+(3)(4)=−1+0+12=11a⋅b=(1)(−1)+(2)(0)+(3)(4)=−1+0+12=11
Next, find the magnitudes of 𝑎a and 𝑏b: ∣𝑎∣=12+22+32=1+4+9=14∣a∣=12+22+32=1+4+9=14 ∣𝑏∣=(−1)2+02+42=1+0+16=17∣b∣=(−1)2+02+42=1+0+16=17
Now, apply the formula for cos𝜃cosθ: cos𝜃=111417≈0.606cosθ=141711≈0.606
The angle 𝜃θ can be calculated using the inverse cosine: 𝜃=cos−1(0.606)≈52.6∘θ=cos−1(0.606)≈52.6∘
Example 2: Determine if Two Lines are Perpendicular
Given Lines: Line 1 with direction vector 𝑎=(3,−3,1)a=(3,−3,1) Line 2 with direction vector 𝑏=(2,2,−2)b=(2,2,−2)
Solution: Calculate the dot product of 𝑎a and 𝑏b: 𝑎⋅𝑏=(3)(2)+(−3)(2)+(1)(−2)=6−6−2=−2a⋅b=(3)(2)+(−3)(2)+(1)(−2)=6−6−2=−2
Since the dot product is not zero, the lines are not perpendicular. If it were zero, that would indicate perpendicularity.
5.Shortest Distance between Two Lines in 3D Geometry
Introduction to Shortest Distance between Two Lines in 3D Geometry
In three-dimensional geometry, finding the shortest distance between two skew lines (lines that do not intersect and are not parallel) is a useful concept, especially in engineering and architecture where you need to ensure structures do not interfere with each other.
Formula for Shortest Distance between Two Skew Lines
The shortest distance 𝑑d between two skew lines can be calculated using the formula involving their direction vectors and a point on each line. Let the lines be represented as:
- Line 1: 𝑟1=𝑎+𝑡𝑏r1=a+tb
- Line 2: 𝑟2=𝑐+𝑠𝑑r2=c+sd
Where 𝑎a and 𝑐c are position vectors of points on the lines, and 𝑏b and 𝑑d are the direction vectors.
The formula for the shortest distance is: 𝑑=∣(𝑐−𝑎)⋅(𝑏×𝑑)∣∣𝑏×𝑑∣d=∣b×d∣∣(c−a)⋅(b×d)∣
Here, 𝑏×𝑑b×d represents the cross product of the direction vectors, which gives a vector perpendicular to both lines. The expression (𝑐−𝑎)(c−a) is the vector connecting a point on Line 1 to a point on Line 2.
Example 1: Calculate the Shortest Distance between Two Skew Lines
Given Lines:
- Line 1: 𝑟1=(1,0,0)+𝑡(1,1,1)r1=(1,0,0)+t(1,1,1)
- Line 2: 𝑟2=(2,3,1)+𝑠(1,−1,2)r2=(2,3,1)+s(1,−1,2)
Solution:
Calculate 𝑏×𝑑b×d: 𝑏=(1,1,1),𝑑=(1,−1,2)b=(1,1,1),d=(1,−1,2) 𝑏×𝑑=∣𝑖𝑗𝑘1111−12∣=(3,−1,−2)b×d=∣∣i11j1−1k12∣∣=(3,−1,−2)
Compute (𝑐−𝑎)⋅(𝑏×𝑑)(c−a)⋅(b×d): 𝑎=(1,0,0),𝑐=(2,3,1)a=(1,0,0),c=(2,3,1) (𝑐−𝑎)=(1,3,1)(c−a)=(1,3,1) (1,3,1)⋅(3,−1,−2)=3−3−2=−2(1,3,1)⋅(3,−1,−2)=3−3−2=−2 ∣(𝑐−𝑎)⋅(𝑏×𝑑)∣=2∣(c−a)⋅(b×d)∣=2
Magnitude of 𝑏×𝑑b×d: ∣𝑏×𝑑∣=32+(−1)2+(−2)2=14∣b×d∣=32+(−1)2+(−2)2=14
Finally, calculate 𝑑d: 𝑑=214≈0.534d=142≈0.534
Example 2: Determine the Shortest Distance between Two Parallel Lines
Though our main focus is skew lines, let's briefly cover parallel lines:
- Line 1: 𝑟1=(1,1,1)+𝑡(2,2,2)r1=(1,1,1)+t(2,2,2)
- Line 2: 𝑟2=(3,3,5)+𝑠(2,2,2)r2=(3,3,5)+s(2,2,2)
Solution: For parallel lines, 𝑏=𝑑b=d, thus 𝑏×𝑑=0b×d=0. The formula simplifies, and we use: 𝑑=∣(𝑐−𝑎)⋅𝑛∣∣𝑛∣d=∣n∣∣(c−a)⋅n∣ where 𝑛n is a normal vector to the direction vector 𝑏b. For simplicity, choose 𝑛=(2,−2,0)n=(2,−2,0), orthogonal to (2,2,2)(2,2,2). (𝑐−𝑎)=(2,2,4)(c−a)=(2,2,4) (𝑐−𝑎)⋅𝑛=2∗2+2∗(−2)+4∗0=0(c−a)⋅n=2∗2+2∗(−2)+4∗0=0 Thus, 𝑑=0d=0 for these specific vectors, which indicates they are the same line or coincide.
Real-World Application
Understanding the shortest distance between lines is crucial in urban planning, where different infrastructure elements must not clash. It's also vital in robotics and automotive industries, where spatial relationships determine the mechanical design and functionality.
Hands-On Activity
Create models using strings or wires to represent the lines. Use paper clips or small weights to mark specific points, and visually assess how changing the orientation of these lines affects their shortest distance. This can help grasp the complex nature of spatial relationships in a tangible way.