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Solve 3-Dimensional Geometry Questions on Planes Instantly

Updated: Aug 27

Welcome to Session 2 on 3-Dimensional Geometry Planes. You can access Session 1 on Planes here.

You have the basic equation of a Plane in place. But, you need to know how and when to apply it to problem solving. Keep reading! You will soon learn how.

You are in Class 12 and it is important for you to learn 3 Dimensional Geometry. You don't have the time to go for extra coaching and you have missed a few classes in school when the topic was being taught.

You can get help in 3 Dimensional Geometry for Class 12 here. In my last blog, you were introduced to the basic equations of a plane. You will now proceed with a few problems testing these concepts.

Initially, you may express some discomfort while understanding Planes. But, believe me, that is short lived!You will start with finding the equation of a plane at a distance of p units from the origin and given the direction cosines of the normal to the plane.

How to find the direction cosines of a normal to the plane.

You'll find the equation of a plane given a point on a plane and the direction ratios of the normal to the plane. These formulas are as shown.


Equation of a plane given a point in the plane and the direction ratios of the normal to the plane.
Equation of a plane

Learn how to find the equation of a plane passing through a point and perpendicular to a plane whose direction ratios are given.

Some of the problems covered in this lesson include How to find the Vector and Cartesian equation of a plane through a point and given a normal vector.

Learn how to find the equation of the plane given the foot of the perpendicular from the origin to the plane.


equation of a plane passing through a point and having a normal vector

3 Dimensional Geometry-Problem 4

Given the equation of a plane, finding the length of the perpendicular from the origin to the plane and the perpendicular distance from the origin is taught in this course. All that you need to do is to shift the constant term to the other side and covert the normal to a unit normal.

In this context, this question teaches you how to find the perpendicular distance from the origin at a glance. The trick is to ensure that the normal is a unit normal.

You are taught how to find the point of intersection between a line and a plane. In this context you'll also learn how to find the image of a point in a plane as an application to the earlier problem.

3 D Geometry-Problem 5

Learn a simple case study when a line is contained in a plane. You need to keep in mind that the line is perpendicular to the normal to the plane.

3-D Geometry Problem 6

Experience finding the distance of a point from a plane measured parallel to a line.

3 D Geometry Problem 7

Explore finding the image of a point in a plane.

See how to find the intercept made by a plane on the coordinate axes. 3 D Geometry problem 8

You simply must hear this out ! Find the equation of a plane containing a point and a line. 3 D Geometry problem 9


Feel the ease with which you can solve Mathematics problems by trying out 3 D Geometry problem 10

How to find the equation of a plane passing through the line of intersection of 2 planes ?

Immerse yourself in the fragrance of 3 D geometry by learning how to calculate the length of the diagonal of a parallelopiped whose planes are parallel to the coordinate planes. This is 3 D Geometry Problem 11 for you.


Learn the simple trick in finding the image of a point in the coordinate planes. This is as simple as finding the reflection of the point in the YZ plane as explained in this problem here. Note that the x coordinate sign changes while the y and z coordinates remain the same.

Watch 3 D Geometry Problem 12 here to understand better.


Next you will move on to a whole set of formulas which are essentially the crux for problem solving in planes. These include:

Finding the equation of a plane passing through 2 points and parallel to a line.


Equation of a plane passing through 2 points and parallel to a given line

Note that the trick to problem solving in Planes is to find the equation of the plane in the Cartesian form as that involves expanding a 3 by 3 determinant in each case. Converting it to the Vector form is then easy!

Finding the equation of a plane passing through a point and parallel to 2 lines.


Equation of a plane passing through a point and parallel to 2 lines

Finding the equation of a plane containing 2 lines.


Formula for equation of a plane containing 2 lines

Finding the equation of a plane passing through 3 points.

How to check if 3 points are coplanar.


Condition of Coplanarity of 3 points

Next, you'll learn about the Intercept form of a plane


intercept form of a plane

Here's a simple formula to find the angle between 2 planes.

Note that to find the acute angle we use the same formula but apply modulus.

Learn this concept through this problem which illustrates how to find the acute angle between 2 planes. Note that you'll need to take the modulus here!

3 D Geometry Problem 13


angle between 2 planes

See how to find the angle between a line and a plane .

This is 3 Dimensional Geometry Problem 14 for you.

How do you check if 2 planes are parallel? Is it just your gut feeling or is there a formula?

Check it out here!


How to check if 2 planes are parallel

Given a plane, how will you find the equation of a plane parallel to this plane. See how to do it here!


Equation of a plane parallel to a given plane

See how to find the perpendicular distance from the origin to a plane


You still feel that some personal help in mathematics may be useful. You can message me for online tutoring.

Most important, enjoy the journey of studying!

I hope this is of use to you. My intention is to serve you better.

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