Understanding 3D Geometry is essential for Class 12 Maths. It helps visualize points, lines, and planes in three-dimensional space. This guide covers fundamental concepts with various equations and properties.
1. Basics of 3D Geometry
The 3-Dimensional Coordinate System consists of three axes: X, Y and Z. A point in space is represented as (x, y, z). The distance between two points 
and is given by:
The concept of direction ratios (DRs) and direction cosines (DCs) of a line help make the study of 3D Geometry simple.
2. Equations of a Line in Space
In Class 11, you have studied Equation of Lines in a plane i. e. 2-Dimensional Geometry. And, there you used the concept of slope of a line.
But here, in 3D Geometry, a line can be uniquely determined under these two conditions:
- A line passing through a point and has a given direction
- A line passing through two given points
For example, a line passing through and parallel to a vector or another line having direction ratios is given in cartesian form as following:
The different forms of a line that we study in Class 12 are:
- Vector Form
- Cartesian Form
- Parametric Form
Understand the various equations of line in the attached pdf at the end.
3. Angle Between Two Lines in 3D Space
If two lines in 3D Geometry have direction ratios and , then the angle θ between them is given by:
4. Foot of the Perpendicular
The foot of the perpendicular from a point to a given line can be found using parametric equations of a line as explained in the attached pdf.
Solve for λ to get the coordinates of the foot of the perpendicular.
To solve such questions, you need to apply the concept of perpendicularity of two lines along with direction ratios.
5. Image of a Point with Respect to a Line
The image of a point in a given line is the reflection of the point across the line. It is found using the midpoint formula and solving parametric equations.
6. Coplanar and Skew Lines in 3D Geometry
Coplanar Lines: Lines that lie in the same plane.
Skew Lines: Lines that are neither parallel nor intersecting.
Two Lines in 3D Geometry, given by:
and
are coplanar if:
7. Point of Intersection of 2 Lines in 3 Dimensional Geometry
Two lines intersect if they satisfy the same set of parametric equations for a particular point. Solve for λ and μ, using any two sets of equations and substitute the values so obtained in the 3rd set of equation, to check if they satisfy.
If no common values of λ and μ satisfy all three equations simultaneously, the lines do not intersect!
Download the attached pdf at the end to understand the complete approach.
8. Shortest Distance
Let’s consider the following two lines in 3-Dimensional Space:
and
are skew lines.
Then the shortest distance between them is given by:
You can also find it easily if the equations of the lines are given in cartesian form as explained in the attached pdf.
9. Section Formulae in 3D Geometry
As mentioned in the previous section on Vector Algebra also, if a line segment joins and and is divided in m:n ratio, the coordinates of the point of division are given as:
And if we put m = n = 1 in the above coordinates, we will get the Mid-point Formula.
10. Miscellaneous
Other important concepts include:
- Coordinates of a point on different planes.
- Perpendicular distance of a point from a plane
- Directions cosines of a line equally inclined to 3 axes and
- Condition for Coplanarity of two lines in 3D Space.
Conclusion
Mastering these 3D Geometry concepts will help in solving problems efficiently. Keep practicing for better understanding!
Keep learning! See you soon with the next interesting topic i.e. Probability.
Discover more from Maths Better
Subscribe to get the latest posts sent to your email.