# 3D Geometry Class 12 Formulas

3D geometry class 12 formulas explain the vector algebra of three-dimensional geometry. Although these formulas are quite simple, students often seem confused while solving 3-dimensional geometry problems. This happens mostly due to inadequate knowledge of 3D geometry formulas and concepts. Therefore, it is necessary that students have a deep knowledge of 3D geometry class 12 formulas before attempting the problems based on them. The article provides a list of important 3D geometry class 12 formulas, their applications and useful tips to help students memorize them.

### List of 3D Geometry Class 12 Formulas

Students can refer to the list of 3D geometry class 12 formulas provided below:

- The cosines of the angles made by a line with positive directions of the coordinate axes are direction cosines of a line.
- l
^{2}+ m^{2}+ n^{2}= 1 holds true if l, m, n are the direction cosines of a line. - Direction cosines of a line joining two points A(x
_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}) are (x_{2}- x_{1})/AB , (y_{2}- y_{1})/AB , (z_{2}- z_{1})/AB where the value of AB = √[(x_{2}- x_{1})^{2}+ (y_{2}- y_{1})^{2}+ (z_{2}- z_{1})^{2}] by using the distance formula - The angle θ between the planes A
_{1}x + B_{1}y + C_{1}z + D_{1}= 0 and A_{2}x + B_{2}y + C_{2}z + D_{2}= 0 is given by cosθ = | (A_{1}A_{2}+B_{1}B_{2}+ C_{1}C_{2})/√(A_{1}^{2}+B_{1}^{2}+ C_{1}^{2}) √(A_{2}^{2 }+ B_{2}^{2}+ C_{2}^{2})| - Vector equation of a line that passes through two given point whose position vector is \(\overrightarrow a\) and \(\overrightarrow b\) is \(\overrightarrow r = \overrightarrow a +λ\overrightarrow b\)

## Applications of 3D Geometry Class 12 Formulas

Three-dimensional geometry finds its place in various scientific experiments and calculations. Here are some applications of 3D geometry class 12 formulas:

- 3D geometry class 12 formulas are also helpful in representing electric signals, velocity and other physical quantities.
- These formulas are helpful in modeling the basic quantities such as fluid flows in fluid mechanics.

## Tips to Memorize 3D Geometry Class 12 Formulas

The introduction of 3D geometry class 12 formulas can seem overwhelming to students at first due to the complex terms involved in them. Here are some tips that will help students ease the memorization process:

**Read and understand logically: **Students should go through the formula derivations and notes given in the textbook. Reading formula explanations will help students get a better idea about each symbol used in them.

**Refer to Study Resources: **Visualizing formulas is the best way to remember them. There are many explanatory videos available on the internet to understand the logic of formulas. Students can refer to them.

**Formula Revision: **To make revision quick and simple, students can also download formula wallpapers and save them as wallpapers on their mobile phones and laptops.

## 3D Geometry Class 12 Formulas Examples

**Example 1: **What will be the vector equation of the line passing through the points (- 2, 0, 1) and (2, 3, 5).

**Solution 1:** Let x and y be the position vectors of the point P(-2, 0, 1) and Q(2, 3, 5).

Then, x = -2\(\hat i\) +\(\hat k\)

y= 2\(\hat i\) + 3\(\hat j\) +5\(\hat k\)

y - x = 4\(\hat i\) + 3\(\hat j\) +4\(\hat k\)

Let r be the position vector for any point on the line.

Thus, the vector equation of the line is: r = -2\(\hat i\) +\(\hat k\) + λ (4\(\hat i\) + 3\(\hat j\) +4\(\hat k\))

Students can download the printable **Maths Formulas Class 12** sheet from below.

## FAQs on 3D Geometry Class 12 Formulas

### What are the Important 3D Geometry Class 12 Formulas?

The important formulas for 3D geometry class 12 formulas are:

- The cosines of the angles made by a line with positive directions of the coordinate axes are direction cosines of a line.
- l
^{2}+ m^{2}+ n^{2}= 1 holds true if l, m, n are the direction cosines of a line. - Direction cosines of a line joining two points A(x
_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}) are (x_{2}- x_{1})/AB , (y_{2}- y_{1})/AB , (z_{2}- z_{1})/AB where the value of AB = √[(x_{2}- x_{1})^{2}+ (y_{2}- y_{1})^{2}+ (z_{2}- z_{1})^{2}] by using the distance formula - The angle θ between the planes A
_{1}x + B_{1}y + C_{1}z + D_{1}= 0 and A_{2}x + B_{2}y + C_{2}z + D_{2}= 0 is given by cosθ = | (A_{1}A_{2}+B_{1}B_{2}+ C_{1}C_{2})/√(A_{1}^{2}+B_{1}^{2}+ C_{1}^{2}) √(A_{2}^{2 }+ B_{2}^{2}+ C_{2}^{2})| - Vector equation of a line that passes through two given point whose position vector is \(\overrightarrow a\) and \(\overrightarrow b\) is \(\overrightarrow r = \overrightarrow a +λ\overrightarrow b\)

### What are the Basic Formulas in 3D Geometry Class 12 Formulas?

The basic formulas in 3D geometry class 12 formulas cover the direction cosines and direction ratios of a line. Students will learn to find the angle between skew lines and vector equation of a line that passes through the given point.

### What are the Important Formulas Covered in 3D Geometry Class 12 Formulas?

The important formulas covered in 3D geometry class 12 formulas are mentioned in this article along with relevant examples that show the usage of the formula. Students are advised to go through the list provided in this article that will help them in covering the topics of direction cosines and direction ratios of line and the equations of planes in vector form.

### How Many Formulas are There in 3D Geometry Class 12 Formulas?

There are around five major formulas in 3D geometry class 12 formulas that if learned well will help students solve the majority of the problems. Hence, students must go through the list of formulas in this article and also should inculcate the tips given in this article to learn them well and easily.

### How can I Memorize 3D Geometry Class 12 Formulas?

To memorize 3D geometry class 12 formulas students should go through the notes provided in the textbook to have a precise understanding of each of them. In case of any doubt, they should take help from teachers or friends to clear them as soon as possible. Doing so will help students to build a clear understanding of formulas to easily memorize them.

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