Find the best tutors and institutes for Class 12 Tuition
Q4:
Find the equation of the line which passes through the point $(1, 2, 3)$ and is parallel to the vector $3\hat{i} + 2\hat{j} - 2\hat{k}$.
Solution :
Step 1: Establishing the Theoretical Framework
To determine the equation of a line in three-dimensional space, we require two components: a position vector $\vec{a}$ of a point through which the line passes and a direction vector $\vec{b}$ parallel to the line [Per Vector Equation of a Line Theorem].
Given:
- Point $P(x_1, y_1, z_1) = (1, 2, 3)$
- Position vector of the point, $\vec{a} = 1\hat{i} + 2\hat{j} + 3\hat{k}$
- Direction vector, $\vec{b} = 3\hat{i} + 2\hat{j} - 2\hat{k}$
Step 2: Vector Form Derivation
The vector equation of a line passing through point $\vec{a}$ and parallel to vector $\vec{b}$ is defined as:
$$\vec{r} = \vec{a} + \lambda\vec{b}$$
Substituting the given vectors into the general formula [By Vector Algebra Axioms]:
$$\vec{r} = (1\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(3\hat{i} + 2\hat{j} - 2\hat{k})$$
Where $\lambda$ is a scalar parameter representing any real number ($\lambda \in \mathbb{R}$).
Step 3: Cartesian Form Derivation
The Cartesian equation of a line passing through $(x_1, y_1, z_1)$ with direction ratios $(a, b, c)$ is given by:
$$\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$$
Given the point $(1, 2, 3)$ and direction ratios $(3, 2, -2)$, we map these values directly into the formula [By Cartesian Coordinate Transformation]:
$$\frac{x - 1}{3} = \frac{y - 2}{2} = \frac{z - 3}{-2}$$
Step 4: Geometric Visualization
Step 5: Final Representation
The solution can be expressed in either vector or Cartesian form, as both represent the identical locus of points in $\mathbb{R}^3$.
| Form | Equation |
|---|---|
| Vector Form | $\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(3\hat{i} + 2\hat{j} - 2\hat{k})$ |
| Cartesian Form | $\frac{x - 1}{3} = \frac{y - 2}{2} = \frac{z - 3}{-2}$ |
Final Solution: The vector equation is $\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(3\hat{i} + 2\hat{j} - 2\hat{k})$ and the Cartesian equation is $\frac{x - 1}{3} = \frac{y - 2}{2} = \frac{z - 3}{-2}$.
More Questions from Class 12 Mathematics Three Dimensional Geometry EXERCISE 11.2
- Q1: Show that the three lines with direction cosines $\frac{12}{13}, \frac{-3}{13}, \frac{-4}{13}$; $\frac{4}{13}, \frac{12}{13}, \frac{3}{13}$; $\frac{3}{13}, \frac{-4}{13}, \frac{12}{13}$ are mutually perpendicular.
- Q2: Show that the line through the points $(1, -1, 2)$, $(3, 4, -2)$ is perpendicular to the line through the points $(0, 3, 2)$ and $(3, 5, 6)$.
- Q3: Show that the line through the points $(4, 7, 8)$, $(2, 3, 4)$ is parallel to the line through the points $(-1, -2, 1)$, $(1, 2, 5)$.
- Q5: Find the equation of the line in vector and in cartesian form that passes through the point with position vector $2\hat{i} - \hat{j} + 4\hat{k}$ and is in the direction $\hat{i} + 2\hat{j} - \hat{k}$.
- Q6: Find the cartesian equation of the line which passes through the point $(-2, 4, -5)$ and parallel to the line given by $\frac{x+3}{3} = \frac{y-4}{5} = \frac{z+8}{6}$.
- Q7: The cartesian equation of a line is $\frac{x-5}{3} = \frac{y+4}{7} = \frac{z-6}{2}$. Write its vector form.
- Q8(i): Find the angle between the following pairs of lines: (i) $\vec{r} = 2\hat{i} - 5\hat{j} + \hat{k} + \lambda(3\hat{i} + 2\hat{j} + 6\hat{k})$ and $\vec{r} = 7\hat{i} - 6\hat{k} + \mu(\hat{i} + 2\hat{j} + 2\hat{k})$
- Q8(ii): Find the angle between the following pairs of lines: (ii) $\vec{r} = 3\hat{i} + \hat{j} - 2\hat{k} + \lambda(\hat{i} - \hat{j} - 2\hat{k})$ and $\vec{r} = 2\hat{i} - \hat{j} - 56\hat{k} + \mu(3\hat{i} - 5\hat{j} - 4\hat{k})$
- Q9(i): Find the angle between the following pair of lines: (i) $\frac{x-2}{2} = \frac{y-1}{5} = \frac{z+3}{-3}$ and $\frac{x+2}{-1} = \frac{y-4}{8} = \frac{z-5}{4}$
- Q9(ii): Find the angle between the following pair of lines: (ii) $\frac{x}{2} = \frac{y}{2} = \frac{z}{1}$ and $\frac{x-5}{4} = \frac{y-2}{1} = \frac{z-3}{8}$
- Q10: Find the values of $p$ so that the lines $\frac{1-x}{3} = \frac{7y-14}{2p} = \frac{z-3}{2}$ and $\frac{7-7x}{3p} = \frac{y-5}{1} = \frac{6-z}{5}$ are at right angles.
- Q11: Show that the lines $\frac{x-5}{7} = \frac{y+2}{-5} = \frac{z}{1}$ and $\frac{x}{1} = \frac{y}{2} = \frac{z}{3}$ are perpendicular to each other.
- Q12: Find the shortest distance between the lines $\vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k})$ and $\vec{r} = (2\hat{i} - \hat{j} - \hat{k}) + \mu(2\hat{i} + \hat{j} + 2\hat{k})$
- Q13: Find the shortest distance between the lines $\frac{x+1}{7} = \frac{y+1}{-6} = \frac{z+1}{1}$ and $\frac{x-3}{1} = \frac{y-5}{-2} = \frac{z-7}{1}$
- Q14: Find the shortest distance between the lines whose vector equations are $\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - 3\hat{j} + 2\hat{k})$ and $\vec{r} = (4\hat{i} + 5\hat{j} + 6\hat{k}) + \mu(2\hat{i} + 3\hat{j} + \hat{k})$
- Q15: Find the shortest distance between the lines whose vector equations are $\vec{r} = (1-t)\hat{i} + (t-2)\hat{j} + (3-2t)\hat{k}$ and $\vec{r} = (s+1)\hat{i} + (2s-1)\hat{j} - (2s+1)\hat{k}$
CBSE Solutions for Class 12 Mathematics Three Dimensional Geometry
Chapters in CBSE - Class 12 Mathematics
Top Tutors who teach Three Dimensional Geometry
Top Tutor
Nerul Sector 21, Mumbai
10 yrs of
Exp
1 student
163 hrs of classes completed
Online Classes
Student's Home
As an Experienced Physics tutor with a passion for igniting young minds, I've helped numerous students from diverse boards achieve academic excellence and secure coveted spots in top institutions including, IITs: My students have cracked the prestigious Indian Institutes of Technology exam mains/advance. MBBS programs: Many of my students have secured admission through NEET in top medical colleges Boards I Cater To • CBSE • ICSE • ISC • IGCSE • IB Diploma (SL/HL) • A/AS Level • O Level • International Boards My Unique Approach:- Personalized attention: One-to-one sessions tailored to your learning style and pace Doubt-clearing sessions: Encouraging questions and clarifying complex concepts Homework made easy: Transforming assignments into engaging activities Building confidence: Fostering a growth mindset and celebrating small wins Coaching module support: I'll help you solve modules from top coaching classes Empowering Independent Learning:- Problem-solving skills: I'll guide you to develop critical thinking and analytical skills Self-study: You'll learn to approach problems with confidence and solve them independently Proven Track Record:- Consistent success: Students achieving top grades and securing admission to premier institutions NEET excellence: My students consistently outperform expectations Engineering Entrance Excellence: Many of my students have outperformed in JEE MAIN/ADVANCE Beyond academics: Instilling critical thinking, problem-solving, and analytical skills Why Choose Me? Dedicated mentor: I'm committed to your success and well-being Expert guidance: With my B.Tech background and teaching experience, I provide expert insights Results-driven: My students' achievements are my top priority Let's Achieve Greatness Together! If you're ready to unlock your potential and excel in Physics and Math, I'm here to support your journey. Let's work together to achieve your goals and make your academic dreams a reality!
I studied physics with Imran Sir in class 11 and 12. He explains the concepts properly with patience. He also taught a lot of tricks to solve questions faster. His classes are fun and i improved a lot over time studying with him.
Top Tutor
Tarnaka Nagarjuna Nagar, Hyderabad
13 yrs of
Exp
28 students
789 hrs of classes completed
700
per hour
Online Classes
I always feel, class 12 tution is the major step to any student, so it is very important to be attentive with time and energy to be able to put best efforts to achieve desired goals.
Top Tutor
Sarv Suvidha Nagar, Indore
1 yrs of
Exp
3 students
195 hrs of classes completed
350
per hour
Online Classes
I am an AI-enabled Social Science educator with experience teaching Grades 6–10 at Physics Wallah. I specialize in simplifying complex concepts through visual PPT-based learning, storytelling, and engaging teaching methods. With a strong focus on conceptual clarity and student engagement, I also integrate AI tools to create modern, effective learning experiences. My goal is to make Social Science easy, interesting, and relatable for every student.
Very Good experience.
Top Tutor
Zafarpur Mahawan, Karari
10 yrs of
Exp
6 students
287 hrs of classes completed
400
per hour
Online Classes
Tutor's home
I am an experienced, qualified teacher and tutor with over 20 years of experience in teaching accountancy, business studies and English literature in CBSE school. Passionate about communication in English, over the years I have helped thousands of students overcome their stage fright and nervousness in expressing themselves in public. So far, I have worked with Dr. Rizvi Springfield School (Prayagraj) and Global Classroom, Lucknow (branch of CMS) and currently freelancing with Planet Spark as an Expert in Public Speaking & Creating Writing. I specialize in teaching Accountancy and Business Studies to classes 11 and 12, along with Spoken English. With a strong academic background, including a B. Com., M. Com., and an M.Ed., as well as completing a Management Development Program, I bring a wealth of knowledge and expertise to my teaching. With over 22 years of administrative experience and 10 years of teaching, I have had the privilege of holding key roles such as PGT English, Commerce teacher, Headmistress, and Principal. These experiences have helped me develop a profound understanding of education, enabling me to cater to the unique needs of my students effectively. Teaching has been the most enjoyable and fulfilling part of my professional journey. Over the years, I have built strong connections with my students, fostering an environment of trust and growth. My passion for education has been recognized through several accolades, including a prestigious Principal Award in Delhi. I conduct online classes where I follow a structured lesson plan to ensure my sessions are engaging and interactive. I use PPTs and videos to make concepts clear and relatable, catering to the diverse learning styles of my students. I also provide comprehensive notes and content to support my students' learning and revision. These resources are meticulously prepared to align with the curriculum and focus on enhancing students’ understanding and performance. My approach to teaching is rooted in creating a supportive and motivating environment. I am committed to helping my students achieve their academic and personal goals while instilling confidence and a love for learning in them. Teaching is not just my profession; it is my passion and purpose.
Ms.Lubna is the best tutor for accountancy grade 12 cbse. She is extremely kind and top tier teaching skills.
Top Tutor
Malleshpalya, Bangalore
12 yrs of
Exp
2 students
17 hrs of classes completed
200
per hour
Online Classes
Have been taking classes since 2005 in bangalore for physics and maths for xi,neet,jee mains,thanks.
Other Subjects in CBSE - Class 12
Download free CBSE - Class 12 Mathematics Three Dimensional Geometry EXERCISE 11.2 worksheets
Download Now
