Equation of Line in Three Dimensions | Cartesian & vector Equation | Class 12 Math Notes Study Material Download Free PDF

Class 12 Math Chapter Three Dimensional Geometry: Important Topics for CBSE Board Exam Students

Written by Author Neeraj Anand (Director of Anand Classes Jalandhar)
Published by Anand Technical Publishers

Equation of a line is defined as y = mx + c, where c is the y-intercept and m is the slope. Vectors can be defined as a quantity possessing both direction and magnitude. Position vectors simply denote the position or location of a point in the three-dimensional Cartesian system with respect to a reference origin.

Further, we shall study in detail about vectors and Cartesian equation of a line in three dimensions. It is known that we can uniquely determine a line if:

• It passes through a particular point in a specific direction, or
• It passes through two unique points

Let us study each case separately and try to determine the equation of a line in both the given cases.

Table of Contents

• Equation of a Line passing through a point and parallel to a vector
• Equation of a Line passing through two given points

Equation of a Line passing through a point and parallel to a vector

Let us consider that the position vector of the given point be  

$\begin{array}{l}\overrightarrow{a}\end{array}$  with respect to the origin. The line passing through point A is given by l and it is parallel to the vector $\begin{array}{l}\overrightarrow{k}\end{array}$ as shown below. Let us choose any random point R on the line l and its position vector with respect to origin of the rectangular co-ordinate system is given by $\begin{array}{l}\overrightarrow{r}\end{array}$.

Since the line segment, $\begin{array}{l}\overline{AR}\end{array}$  is parallel to vector $\begin{array}{l}\overrightarrow{k}\end{array}$ , therefore for any real number α, $\begin{array}{l}\overline{AR}\end{array}$ = α  $\begin{array}{l}\overrightarrow{k}\end{array}$

Also, 

$\begin{array}{l}\overline{AR}\end{array}$ = $\begin{array}{l}\overline{OR}\end{array}$ –  $\begin{array}{l}\overline{OA}\end{array}$

Therefore, α $\begin{array}{l}\overrightarrow{r}\end{array}$ = $\begin{array}{l}\overrightarrow{r}\end{array}$ – $\begin{array}{l}\overrightarrow{a}\end{array}$

From the above equation it can be seen that for different values of α, the above equations give the position of any arbitrary point R lying on the line passing through point A and parallel to vector k. Therefore, the vector equation of a line passing through a given point and parallel to a given vector is given by:

$\begin{array}{l}\overrightarrow{r}\end{array}$ = $\begin{array}{l}\overrightarrow{a}\end{array}$ + α $\begin{array}{l}\overrightarrow{k}\end{array}$

If the three-dimensional co-ordinates of the point ‘A’ are given as (x1, y1, z1) and the direction cosines of this point is given as a, b, c then considering the rectangular co-ordinates of point R as (x, y, z):

Substituting these values in the vector equation of a line passing through a given point and parallel to a given vector and equating the coefficients of unit vectors i, j and k, we have,

Eliminating α we have:

This gives us the Cartesian equation of line.

Equation of a Line passing through two given points

Let us consider that the position vector of the two given points A and B be $\begin{array}{l}\overrightarrow{a}\end{array}$ and $\begin{array}{l}\overrightarrow{b}\end{array}$  with respect to the origin. Let us choose any random point R on the line and its position vector with respect to origin of the rectangular co-ordinate system is given by $\begin{array}{l}\overrightarrow{r}\end{array}$.

Point R lies on the line AB if and only if the vectors $\begin{array}{l}\overline{AR}\end{array}$ and $\begin{array}{l}\overline{AB}\end{array}$  are collinear. Also, $\begin{array}{l}\overline{AR}\end{array}$ =  $\begin{array}{l}\overrightarrow{r}\end{array}$ – $\begin{array}{l}\overrightarrow{a}\end{array}$ $\begin{array}{l}\overline{AB}\end{array}$ =  $\begin{array}{l}\overrightarrow{b}\end{array}$ – $\begin{array}{l}\overrightarrow{a}\end{array}$

Thus R lies on AB only if;

$\begin{array}{l}\overrightarrow{r}–\overrightarrow{a}=\alpha (\overrightarrow{b}–\overrightarrow{a})\end{array}$

Here α is any real number.
From the above equation it can be seen that for different values of α, the above equation gives the position of any arbitrary point R lying on the line passing through point A and B. Therefore, the vector equation of a line passing through two given points is given by:

$\begin{array}{l}\overrightarrow{r}=\overrightarrow{a}+\alpha (\overrightarrow{b}–\overrightarrow{a})\end{array}$

If the three-dimensional coordinates of the points A and B are given as (x1, y1, z1) and (x2, y2, z2) then considering the rectangular co-ordinates of point R as (x, y, z)

Substituting these values in the vector equation of a line passing through two given points and equating the coefficients of unit vectors i, j and k, we have

Eliminating α we have:

This gives us the Cartesian equation of a line.

For CBSE Class 12 students, the chapter on Three Dimensional Geometry is crucial in both board exams and competitive tests. This chapter introduces students to a world beyond two dimensions and equips them with the tools to understand 3D shapes, positions, and relationships. Neeraj Anand, the esteemed Director of Anand Classes, Jalandhar, has carefully crafted a detailed guide to help students excel in this topic. Published by Anand Technical Publishers, this book simplifies complex concepts and ensures thorough understanding.

Key Topics in Class 12 Math Three Dimensional Geometry for CBSE:

1. Introduction to 3D Geometry: Understanding the basic concepts of three-dimensional space, coordinates, and the coordinate axes.
2. Direction Ratios and Direction Cosines: Mastering the concepts of direction ratios and direction cosines to describe the orientation of lines in 3D space.
3. Equation of a Line in 3D: Learn how to derive and work with the equation of a line in space using both parametric and symmetric forms.
4. Equation of a Plane: Understand the equation of a plane, including the general form and the plane passing through a point and normal to a vector.
5. Shortest Distance between Two Skew Lines: Learn how to find the shortest distance between two non-intersecting and non-parallel lines.
6. Angle between Two Lines and Planes: Master how to calculate the angle between two lines or between a line and a plane in 3D space.
7. Distance from a Point to a Plane: Learn how to determine the perpendicular distance from a point to a given plane.

Why This Book is Essential:

• Comprehensive Coverage: The book covers all important concepts, definitions, and formulas, ensuring that students have a solid understanding of Three Dimensional Geometry.
• Detailed Problem Solving: With a variety of problems and detailed solutions, this book helps students master the techniques required to solve even the most challenging questions.
• Student-Friendly Approach: Neeraj Anand’s teaching style focuses on simplifying complex topics, making them easy to understand and apply.
• CBSE Syllabus Aligned: The content is specifically designed to align with the latest CBSE syllabus, so students can be confident they are thoroughly prepared.

Perfect for CBSE Board Exam Preparation:

Three Dimensional Geometry can be intimidating, but with the right guidance, it can be mastered. Neeraj Anand’s book provides that guidance, ensuring that students have the tools they need to tackle this challenging chapter with confidence and achieve high marks in the CBSE Board Exams.

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Don’t wait! Grab your copy of Class 12 Math Chapter Three Dimensional Geometry by Neeraj Anand today, available at all leading bookstores or through Anand Technical Publishers.

With Neeraj Anand’s expert insights, you’ll not only grasp the intricacies of Three Dimensional Geometry but also build a strong foundation for future studies in mathematics, physics, and engineering.

Let’s make Three Dimensional Geometry your strongest asset for the CBSE Board Exams! 🌟