Updated: Nov 26
How to study 3D Geometry Class 12?
Welcome to three dimensional Geometry for Class 12. I am Suman Mathews, mathematics teacher and content developer with a teaching experience of three decades. Students in general tend to find this topic difficult. I hope I can help eliminate this difficutly for you. Following are the topics which I will be teaching in my online classes along with lots and lots of problem solving.
You'll also use concepts learnt in this chapter in CUET Mathematics.
You'll start with by learning the direction ratios and direction cosines of a line. The direction ratios are the angles made by a line with the positive directions of the x,y and z axes respectively. The direction cosines are the cosines of these angles.
Learn the formulas for finding the direction ratios and direction cosines of a line given two points on a line. Using this, you can also find the angle between the two lines. Note that when you take the modulus, you'll get the acute angle between the two lines. If the cosine of the angle between the lines is zero, the lines are perpendicular. Yet another important point to keep in mind is that when two lines are parallel, their direction ratios are proportional.
Understanding Direction Cosines in 3 Dimensional Geometry
MCQ-Direction Cosines in 3 d Geometry
How to calculate equation of a line in three dimensional geometry?
Moving on to equation of a line in three dimensional geometry. You need to learn how to write the equation of a line in Cartesian and Vector form. Basically, you should be able to convert one form to the other.
You'll need to learn how to calculate the equation of a line passing through a given point and parallel to a given vector. Equation of a line passing through two points is also what you'll be learning.The Cartesian form of a line is slightly easier to calculate and then you can always convert to the Vector form.
Tips on finding equation of a line in 3 D Geometry
An important aspect of three dimensional Geometry that you'll need to know is finding the angle between two lines. This can be done using the vector form or Cartesian form of a line. Note that when you apply the modulus to the formula, it's finding the acute angle between two lines.
You'll also need to know how to find the equation of a line passing through a point and perpendicular to two lines.
Free Video on how to find the angle between 2 lines in 3 D Geometry
Next, you'll need to learn how to calculate the shortest distance between two lines. The vector form of the equation is easier to calculate in this case. Keep in mind that two non parallel lines intersect if and only if the shortest distance between them is zero. You can also calculate the shortest distance between two parallel lines as an application of this formula.
A plane is a surface such that if any two distinct points are taken on it, then the line containing these points lies completely in it. You'll start with calculating the equation of a plane perpendicular to a given direction and at a distance p from the origin. This is the normal form and all other equations can be derived from this.
How to calculate equation of a plane in three dimensional geometry?
How to find the equation of a plane
MCQ-Find the length of the perpendicular from the origin to the plane
You'll also learn how to calculate the equation of a plane perpendicular to a given direction and passing through a given point. Again, you can write the equation of the plane in the Cartesian and Vector form. You should know how to derive one form from the other. All these problems involve identifying the direction ratios of the normal to the plane.
Finding the direction cosines of the normal to a plane-MCQ
MCQ-Finding the image of a point in a plane
The next section is extremely important as it covers various points extremely important in three dimensional geometry. These include
Finding the equation of a plane passing through two given points and parallel to a given line.
Finding the equation of a plane passing through a point and parallel to two non parallel lines.
Finding the condition of coplanarity of two lines.
Finding the equation of a plane containing two lines.
Finding the equation of a plane passing through three points.
Intercept form of a plane.
Finding the equation of a plane passing through the intersection of two planes. This is essentially the most important part of planes and you'll need to have a good understanding of these. I would personally recommend using the Cartesian form to calculate each of the above. You simply have to reduce everything to a 3 by 3 determinant.
How to find the equation of a plane-lesson
Equation of a plane -Question
Question-Find the image of a point in a plane
Question- Find the equation of a plane containing a point and a line
Question- Find the equation of a plane containing the point of intersection of two planes
MCQ-Equation of a plane
MCQ-Equation of a plane containing a line
MCQ-Intercept form of a plane
Moving on, you'll learn how to calculate the angle between two planes. Calculating the angle between a line and a plane is also important and something that you have to know.
How to find the angle between two planes
MCQ-Find the acute angle between two planes
MCQ-Find the angle between a line and a plane
How to calculate distance of a point from a plane?
How to calculate the distance of a point from a plane is what you'll learn next. That means you'll calculate the length of the perpendicular from a point to a plane. As I mentioned earlier, it's easier to remember the Cartesian form of the formula here. To calculate the perpendicular distance between two parallel planes, take any point on one plane and find the perpendicular distance from this point to the other plane.
So, this is 3D Geometry Class 12
How to find the distance of a point from a plane
How to find the distance of a point from a plane measured parallel to a line-
Question- Find the distance between two parallel planes
Question on Three dimensional geometry
Do you still need help? Join my online classes!
Learn all this and more by registering for my online classes. You'll learn multiple choice questions, Assertion based questions, case study based questions, conceptual based questions and more. There will also be tests at the end of every chapter.
You can sign up for questions on Three dimensional geometry. So, let's start the learning process.