Find the best tutors and institutes for Class 12 Tuition
Vector Algebra EXERCISE 10.3 Solutions
- Q1: Find the angle between two vectors $\vec{a}$ and $\vec{b}$ with magnitudes $\sqrt{3}$ and 2, respectively having $\vec{a} \cdot \vec{b} = \sqrt{6}$.
- Q2: Find the angle between the vectors $\hat{i} - 2\hat{j} + 3\hat{k}$ and $3\hat{i} - 2\hat{j} + \hat{k}$
- Q3: Find the projection of the vector $\hat{i} - \hat{j}$ on the vector $\hat{i} + \hat{j}$.
- Q4: Find the projection of the vector $\hat{i} + 3\hat{j} + 7\hat{k}$ on the vector $7\hat{i} - \hat{j} + 8\hat{k}$.
- Q5: Show that each of the given three vectors is a unit vector: $\frac{1}{7}(2\hat{i} + 3\hat{j} + 6\hat{k})$, $\frac{1}{7}(3\hat{i} - 6\hat{j} + 2\hat{k})$, $\frac{1}{7}(6\hat{i} + 2\hat{j} - 3\hat{k})$ Also, show that they are mutually perpendicular to each other.
- Q6: Find $|\vec{a}|$ and $|\vec{b}|$, if $(\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = 8$ and $|\vec{a}| = 8|\vec{b}|$.
- Q7: Evaluate the product $(3\vec{a} - 5\vec{b}) \cdot (2\vec{a} + 7\vec{b})$.
- Q8: Find the magnitude of two vectors $\vec{a}$ and $\vec{b}$, having the same magnitude and such that the angle between them is $60^{\circ}$ and their scalar product is $\frac{1}{2}$.
- Q9: Find $|\vec{x}|$, if for a unit vector $\vec{a}$, $(\vec{x} - \vec{a}) \cdot (\vec{x} + \vec{a}) = 12$.
- Q10: If $\vec{a} = 2\hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{b} = -\hat{i} + 2\hat{j} + \hat{k}$ and $\vec{c} = 3\hat{i} + \hat{j}$ are such that $\vec{a} + \lambda\vec{b}$ is perpendicular to $\vec{c}$, then find the value of $\lambda$.
- Q11: Show that $|\vec{a}|\vec{b} + |\vec{b}|\vec{a}$ is perpendicular to $|\vec{a}|\vec{b} - |\vec{b}|\vec{a}$, for any two nonzero vectors $\vec{a}$ and $\vec{b}$.
- Q12: If $\vec{a} \cdot \vec{a} = 0$ and $\vec{a} \cdot \vec{b} = 0$, then what can be concluded about the vector $\vec{b}$?
- Q13: If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors such that $\vec{a} + \vec{b} + \vec{c} = \vec{0}$, find the value of $\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}$.
- Q14: If either vector $\vec{a} = \vec{0}$ or $\vec{b} = \vec{0}$, then $\vec{a} \cdot \vec{b} = 0$. But the converse need not be true. Justify your answer with an example.
- Q15: If the vertices A, B, C of a triangle ABC are (1, 2, 3), (–1, 0, 0), (0, 1, 2), respectively, then find $\angle ABC$. [$\angle ABC$ is the angle between the vectors $\vec{BA}$ and $\vec{BC}$].
- Q16: Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, –1) are collinear.
- Q17: Show that the vectors $2\hat{i} - \hat{j} + \hat{k}$, $\hat{i} - 3\hat{j} - 5\hat{k}$ and $3\hat{i} - 4\hat{j} - 4\hat{k}$ form the vertices of a right angled triangle.
- Q18: If $\vec{a}$ is a nonzero vector of magnitude ‘a’ and $\lambda$ a nonzero scalar, then $\lambda\vec{a}$ is unit vector if
Other Exercises for CBSE - Class 12 Mathematics Vector Algebra
Chapters in CBSE - Class 12 Mathematics
Top Tutors who teach Vector Algebra
Top Tutor 
Yamare Village, Bangalore 
8 yrs of Exp 
1 student 
8 hrs of classes completed 
1500 per hour 
Student's Home 
Tutor's home Hello! I’m Anwar, a certified mathematics tutor with over 10 years of experience helping students excel at every stage, from Primary, Middle School to High School and University level. My passion is to make math clear, engaging, and enjoyable for learners worldwide. I specialize in teaching Maths to students from Grade 8 to Grade 12 across multiple curricula including IGCSE/Cambridge, IB (MYP), GCSE, CBSE and ICSE. I focus on concept clarity and exam-oriented preparation, helping students understand not just how to solve questions, but how to approach them logically. My teaching approach is highly interactive and student-centric. Each session begins with clear explanation of concepts, followed by strategies to decode questions and step-by-step solving methods. Students actively participate during the class and solve questions along with me, which allows me to check their understanding in real time. Once the concepts are tested, students solve questions independently while I closely monitor their working and provide instant feedback wherever required. This method helps students strengthen accuracy, speed and confidence, and is especially effective for students who feel weak or anxious in Maths and Science. I offer one-to-one personalized tutoring with a minimum class duration of one hour, ensuring complete attention and customized learning for every student.
Top Tutor Rahara Kalyan Nagar, Kolkata 17 yrs of Exp 2 students 249 hrs of classes completed 600 per hour 
Online Classes Student's Home Tutor's home Given Private tuition to students preparing for CBSE or WBCHSE board exams. Guided students preparing for different competitive exams, and got satisfactory results and feedback at the end.
Top Tutor Begur Road, Bangalore 5 yrs of Exp 4 hrs of classes completed 350 per hour Online Classes Tutor's home I have been teaching since last more than 5 years. My way of teaching style is so simple so that it make students very easier to understand. I am taking classes of Math and Hindi subject.
Top Tutor HSR Layout, Bangalore 20 yrs of Exp 30 students 1734 hrs of classes completed 700 per hour Online Classes Tutor's home With a distinguished Doctorate in Chemistry, I bring 28 years of expertise, seamlessly integrating profound knowledge and unparalleled teaching prowess. As a Chemistry Teacher for CBSE and IBDP Chemistry Faculty, I've significantly contributed to intellectual growth in prestigious institutions like Kendriya Vidyalaya, Ecole Mondiale World School, and Singapore International School etc. Recognized as a trailblazer, I've served as an Assistant Examiner and Moderator for Internal Assessment in IBDP Chemistry for six years, upholding the highest evaluation standards and showcasing a deep understanding of curriculum intricacies. My extensive experiences beyond conventional teaching establish me as a luminary in Chemistry education, leaving an enduring legacy in the annals of education. I remain dedicated to inspiring future generations to achieve unparalleled heights in scientific inquiry and knowledge.
A very knowledgeable and caring teacher Patient and passionate teaching. All these qualities helped me succeed and score well in my board exams.
Top Tutor Kharghar, Mumbai 20 yrs of Exp 1 student 179 hrs of classes completed 500 per hour Online Classes Tutor's home I have over 22 years of teaching experience in reputed schools in India, with extensive experience in teaching Physics and Science at the Secondary and Senior Secondary levels. I have taught the CBSE/NCERT curriculum for Classes 9–12, with a strong focus on Physics for Classes 11 and 12 and board examination preparation. I served as the Head of the Science Department (HOD) for 7 years, providing academic leadership, curriculum planning, teacher coordination and mentoring. I have also worked as a Head Examiner for evaluation work, giving me valuable experience in assessment standards, question-paper analysis and examination evaluation. My students have consistently achieved proven and strong board examination results, reflecting my focus on conceptual clarity, systematic preparation, problem-solving and competency-based learning. My teaching methodology combines rigorous academic preparation with real-life applications and the effective use of digital teaching tools. I hold an M.Sc. in Physics and B.Ed., along with a PGDCA. I am a Microsoft Certified Educator and have completed professional certifications in online teaching and Design Thinking. In 2018, I secured 3rd rank in the CENTA Teaching Professionals’ Olympiad. Over the years, I have taught students with diverse learning abilities and developed a student-centred approach focused on conceptual understanding, confidence-building and excellent examination performance. I currently offer both online and offline individual and group classes.
Really happy with the Physics classes! Tuli Maam explains concepts clearly and makes even difficult topics easy to understand. I really appreciate that she focuses on making students understand the concepts rather than simply memorising formulas. Her guidance has helped build confidence and develop a much better understanding of Physics. Highly recommended!
