Three Dimensional Geometry: Chapter 11 Links
Current page: Three Dimensional Geometry 11.1 NCERT Solutions. Go to other exercises and content from this chapter :
Three Dimensional Geometry Exercise 11.1 introduces the basic concepts of direction cosines and direction ratios of a line. In this exercise, you will learn how to find the direction cosines of a line, determine the direction ratios, obtain them from two given points, and use them to check whether three points are collinear.
\displaystyle l=\cos\alpha,\qquad m=\cos\beta,\qquad n=\cos\gamma,\qquad l^2+m^2+n^2=1
The main idea is to understand the direction of a line in three-dimensional space. You will also learn the relationship between direction cosines and direction ratios.
By practising these questions, you will build a strong foundation in Three Dimensional Geometry, making it easier to study the equations of lines and other concepts in the upcoming exercises. These concepts are very important for the CBSE board examinations and CUET.
Key Concepts
Before solving Exercise 11.1, it is important to understand the concepts of direction cosines and direction ratios of a line. These ideas form the foundation for studying lines and planes in three-dimensional geometry.
1. Direction Cosines (d.c.’s)
If a directed line makes angles \alpha,\beta,\gamma with the positive x, y and z axes respectively, then
\displaystyle l=\cos\alpha,\qquad m=\cos\beta,\qquad n=\cos\gamma
Here, l, m and n are called the direction cosines of the line.
They satisfy
\displaystyle l^2+m^2+n^2=1
2. Direction Ratios (d.r.’s)
Any three numbers proportional to the direction cosines of a line are called its direction ratios.
If a,b,c are the direction ratios, then the corresponding direction cosines are
\displaystyle l=\pm\frac{a}{\sqrt{a^2+b^2+c^2}},\quad m=\pm\frac{b}{\sqrt{a^2+b^2+c^2}},\quad n=\pm\frac{c}{\sqrt{a^2+b^2+c^2}}
In other words, if (a,b,c) are the direction ratios and (l,m,n) are the corresponding direction cosines, then
\displaystyle l=\frac{a}{r},\qquad m=\frac{b}{r},\qquad n=\frac{c}{r},\qquad r=\sqrt{a^2+b^2+c^2}
or \displaystyle a=lr,\qquad b=mr,\qquad c=nr
3. Direction Ratios of a Line Passing Through Two Points
If a line passes through the points P(x_1,y_1,z_1) and Q(x_2,y_2,z_2), then one set of its direction ratios is
\displaystyle x_2-x_1,\qquad y_2-y_1,\qquad z_2-z_1
Another valid set is
\displaystyle x_1-x_2,\qquad y_1-y_2,\qquad z_1-z_2
4. Direction Cosines of a Line Passing Through Two Points
If PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}, then the direction cosines are
\displaystyle \left(\frac{x_2-x_1}{PQ},\;\frac{y_2-y_1}{PQ},\;\frac{z_2-z_1}{PQ}\right)
5. Direction Cosines and Direction Ratios of the Coordinate Axes
- x-axis: Direction cosines are (1,0,0), Direction ratios are (1,0,0)
- y-axis: Direction cosines are (0,1,0), Direction ratios are (0,1,0)
- z-axis: Direction cosines are (0,0,1), Direction ratios are (0,0,1)
6. Important Points
- A line has infinitely many sets of direction ratios, but only two sets of direction cosines (equal in magnitude and opposite in sign).
- Direction ratios are always proportional to the direction cosines.
- Direction cosines always satisfy l^2+m^2+n^2=1.
- To check whether three points are collinear, compare the direction ratios of the two line segments. If two lines are parallel (or collinear), then their direction ratios are the same or proportional. Conversely, if the direction ratios of two lines are proportional, the lines are parallel (or collinear if they have a common point).
Quick Reference Table
| Concept | Details |
|---|---|
| Direction Cosines (d.c.’s) | \displaystyle l=\cos\alpha;\; m=\cos\beta;\; n=\cos\gamma;\; l^2+m^2+n^2=1 |
| Direction Ratios (d.r.’s) |
Any three numbers proportional to the direction cosines of a line. \displaystyle a=lr;\; b=mr,\; c=nr;\; r=\sqrt{a^2+b^2+c^2} |
| Line Through Two Points |
If the line passes through P(x_1,y_1,z_1) and Q(x_2,y_2,z_2), then one set of direction ratios is \displaystyle (x_2-x_1,\;y_2-y_1,\;z_2-z_1) |
| Direction Cosines from Two Points |
\displaystyle \left(\frac{x_2-x_1}{PQ},\;\frac{y_2-y_1}{PQ},\;\frac{z_2-z_1}{PQ}\right)
where PQ is the distance between P and Q. |
| Coordinate Axes |
x-axis: (1,0,0), y-axis: (0,1,0), z-axis: (0,0,1) |
Before You Begin… Remember that direction ratios need not satisfy any condition, but direction cosines always satisfy l^2+m^2+n^2=1. Do not confuse the two.
Let us now solve all the NCERT questions of Three Dimensional Geometry 11.1 step-by-step.
Question 1: Direction Cosines of a Line
1. If a line makes angles 90°, 135° and 45° with the x, y and z-axes respectively, find its direction cosines.
Solution
Think First… The direction cosines of a line are simply the cosines of the angles made by the line with the positive x, y and z-axes. So, find cos 90°, cos 135° and cos 45°.
Let the direction cosines of the line be l, m and n.
We know that
\displaystyle l=\cos\alpha,\qquad m=\cos\beta,\qquad n=\cos\gamma
Here,
- \displaystyle \alpha=90^\circ
- \displaystyle \beta=135^\circ
- \displaystyle \gamma=45^\circ
Therefore,
\displaystyle l=\cos90^\circ=0;\; m=\cos135^\circ=-\frac{1}{\sqrt2};\; n=\cos45^\circ=\frac{1}{\sqrt2}
Hence, the direction cosines of the given line are
\boxed{\displaystyle 0,-\frac{1}{\sqrt2},\frac{1}{\sqrt2}}
Question 2: Three Dimensional Geometry 11.1
2. Find the direction cosines of a line which makes equal angles with the coordinate axes.
Solution
Think First… If a line makes equal angles with the x, y and z-axes, then all its direction cosines are equal. Use the relation l^2+m^2+n^2=1 to find their value.
Since the line makes equal angles with the coordinate axes,
\displaystyle \therefore l=m=n
Also, we know that
\displaystyle l^2+m^2+n^2=1
Substituting l=m=n, we get
\displaystyle 3l^2=1
\displaystyle \Rightarrow l=\pm\frac{1}{\sqrt3}
Since l=m=n,
Therefore, \displaystyle l=m=n=\pm\frac{1}{\sqrt3}
Hence, the direction cosines of the given line are
\boxed{\displaystyle \pm\frac{1}{\sqrt3},\,\pm\frac{1}{\sqrt3},\,\pm\frac{1}{\sqrt3}}
You may already be following Maths Better for the NCERT Solutions of the earlier Class 12 Maths chapters. If not, do check them out after completing this exercise.
Likewise, Three Dimensional Geometry 11.1 lays the foundation of the entire chapter by introducing direction cosines and direction ratios. Once you clearly understand how to obtain them, relate them to each other, and use them to determine the direction of a line and check the collinearity of points, the remaining exercises will become much easier. So, take your time to master these fundamentals before moving ahead. Now, let’s proceed to the next question.
Question 3: 3-D Geometry
3. If a line has the direction ratios −18, 12, −4, then what are its direction cosines?
Solution
Think First… To find the direction cosines from the direction ratios, divide each direction ratio by its magnitude \sqrt{a^2+b^2+c^2}.
The given direction ratios are
\displaystyle a=-18,\qquad b=12,\qquad c=-4
The magnitude of the direction ratio vector is given by
\displaystyle r=\sqrt{a^2+b^2+c^2}
Substituting the values, we get
\displaystyle r=\sqrt{(-18)^2+12^2+(-4)^2}
\displaystyle =\sqrt{324+144+16}=\sqrt{484}=22
Therefore,
\displaystyle l=\frac{a}{r}=\frac{-18}{22}=-\frac{9}{11}
\displaystyle m=\frac{b}{r}=\frac{12}{22}=\frac{6}{11}
and \displaystyle n=\frac{c}{r}=\frac{-4}{22}=-\frac{2}{11}
Hence, the direction cosines of the given line are
\boxed{\displaystyle -\frac{9}{11},\frac{6}{11},-\frac{2}{11}}
Question 4: Three Dimensional Geometry 11.1
5. Show that the points (−2, 4, 5), (1, 2, 3) and (7, −2, −1) are collinear.
Solution
Think First… Find the direction ratios of two line segments having a common point. If the direction ratios are the same or proportional, then the three points are collinear.
Let the given points be
\displaystyle A(-2,4,5),\qquad B(1,2,3),\qquad C(7,-2,-1)
So, the direction ratios of the line joining A and B are
\displaystyle 1-(-2),\quad 2-4,\quad 3-5
i.e. \displaystyle 3,\,-2,\,-2
Also, the direction ratios of the line joining B and C are
\displaystyle 7-1,\quad -2-2,\quad -1-3
i.e. \displaystyle 6,\,-4,\,-4
Since, the ratios of the corresponding d.r.’s is
\displaystyle \frac{6}{3}=\frac{-4}{-2}=\frac{-4}{-2}=2
i.e. the direction ratios are proportional. Hence, AB is parallel to BC.
Since B is common to both the line segments, the points A, B and C are collinear.
Hence proved.
By now, you have already completed the NCERT Solutions for Matrices, Determinants, Relations and Functions, Inverse Trigonometric Functions, Continuity and Differentiability, Application of Derivatives, Integrals, Application of Integrals, Differential Equations and Vector Algebra.
Similarly, every exercise of Three Dimensional Geometry is explained here with detailed, step-by-step solutions to help you develop a clear understanding of 3-D Geometry concepts and prepare confidently for your CBSE board examinations.
Many of these questions are also available in video format on my YouTube Channel, @MathsBetter, where each solution is explained in a simple and student-friendly manner. Now, let’s proceed to the last question of Exercise 11.1.
Question 5: Three Dimensional Geometry 11.1
5. Find the direction cosines of the sides of the triangle whose vertices are (3, 5, −4), (−1, 1, 2) and (−5, −5, −2).
Solution
Think First… To find the direction cosines of a side, first find its direction ratios, then divide each by the magnitude of the direction ratio vector.
Let
\displaystyle A(3,5,-4),\qquad B(-1,1,2),\qquad C(-5,-5,-2)
Direction cosines of side AB
The direction ratios of AB are
\displaystyle -1-3,\;1-5,\;2-(-4)
or \displaystyle -4,-4,6
Its magnitude is
\displaystyle \sqrt{(-4)^2+(-4)^2+6^2}=\sqrt{68}=2\sqrt{17}
Hence, the direction cosines of AB are
\boxed{\displaystyle -\frac{2}{\sqrt{17}},-\frac{2}{\sqrt{17}},\frac{3}{\sqrt{17}}}
Direction cosines of side BC
The direction ratios of BC are
\displaystyle -5-(-1),\;-5-1,\;-2-2
or \displaystyle -4,-6,-4
Its magnitude is
\displaystyle \sqrt{(-4)^2+(-6)^2+(-4)^2}=\sqrt{68}=2\sqrt{17}
Hence, the direction cosines of BC are
\boxed{\displaystyle -\frac{2}{\sqrt{17}},-\frac{3}{\sqrt{17}},-\frac{2}{\sqrt{17}}}
Direction cosines of side CA
The direction ratios of CA are
\displaystyle 3-(-5),\;5-(-5),\;-4-(-2)
or \displaystyle 8,10,-2
Its magnitude is
\displaystyle \sqrt{8^2+10^2+(-2)^2}=\sqrt{168}=2\sqrt{42}
Hence, the direction cosines of CA are
\boxed{\displaystyle \frac{4}{\sqrt{42}},\frac{5}{\sqrt{42}},-\frac{1}{\sqrt{42}}}
Maths Better Tip! Direction Ratios vs Direction Cosines: Direction ratios can be any three numbers proportional to one another, whereas direction cosines are obtained by dividing the direction ratios by their magnitude. Therefore, direction cosines always satisfy l^2+m^2+n^2=1, while direction ratios need not. Keeping this distinction clear will help you avoid one of the most common mistakes in Three Dimensional Geometry.
Common Mistakes to Avoid
- Confusing direction cosines with direction ratios: Direction cosines satisfy l^2+m^2+n^2=1, whereas direction ratios need not satisfy this condition.
- Using the wrong sign: The signs of the direction cosines depend on the angles made with the positive coordinate axes. Always check the quadrant before writing the values.
- Not simplifying direction ratios: Any proportional set of numbers represents the same direction ratios. Simplify them whenever possible.
- Finding direction cosines incorrectly: Divide each direction ratio by \sqrt{a^2+b^2+c^2}, not by their sum.
- Ignoring the opposite direction of a line: A line has two opposite directions, so its direction cosines may differ only in sign.
- Using the coordinates of the points incorrectly: While finding direction ratios, subtract the coordinates in the same order for all three coordinates.
- Making arithmetic mistakes: Be careful while calculating \sqrt{a^2+b^2+c^2}. An error here affects all three direction cosines.
- Checking collinearity incorrectly: Three points are collinear only if the direction ratios of the two line segments are the same or proportional.
- Using the wrong direction for a side: Reversing the order of the points changes the signs of the direction ratios and direction cosines, but both represent the same line.
- Forgetting that direction cosines are cosines: Always verify that l^2+m^2+n^2=1. If not, the values cannot be direction cosines.
Continue Learning
In Three Dimensional Geometry 11.1, we learned the concepts of direction cosines and direction ratios of a line. We also found direction cosines from the given direction ratios, obtained them using the coordinates of two points, and used direction ratios to check whether three points are collinear.
Before starting Exercise 11.2, revise the following concepts:
- The meaning of direction cosines and the relation l^2+m^2+n^2=1.
- The difference between direction cosines and direction ratios, and how they are related.
- How to find the direction ratios of a line passing through two given points.
- How to obtain the direction cosines from the given direction ratios by dividing each ratio by \sqrt{a^2+b^2+c^2}.
- How to determine the direction cosines of the sides of a triangle using the coordinates of its vertices.
- How to check whether three points are collinear by comparing the direction ratios of two line segments.
- The direction cosines and direction ratios of the x-, y– and z-axes.
- The fact that reversing the direction of a line changes the signs of its direction cosines and direction ratios, but not the line itself.
Explore More
A good understanding of direction cosines and direction ratios is essential for mastering Three Dimensional Geometry. Learn how they are related, how to obtain them from the coordinates of two points, and how to use them to check whether three points are collinear. Always remember that direction cosines satisfy l^2+m^2+n^2=1, whereas direction ratios need not. With these concepts clear, you will find the upcoming exercises on the equations of lines and planes much easier to understand and solve.
All the best and keep learning 👍
