Find the best tutors and institutes for Class 12 Tuition
Q16:
If $\theta$ is the angle between two vectors $\vec{a}$ and $\vec{b}$, then $\vec{a} \cdot \vec{b} \ge 0$ only when
Solution :
Step 1: Definition of Scalar (Dot) Product
The scalar product (dot product) of two non-zero vectors $\vec{a}$ and $\vec{b}$ is defined by the geometric relation: $$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta$$ where $|\vec{a}|$ and $|\vec{b}|$ are the magnitudes of vectors $\vec{a}$ and $\vec{b}$ respectively, and $\theta$ is the angle between them such that $0 \le \theta \le \pi$. [Per Definition of Scalar Product in Euclidean space].
Step 2: Analysis of Magnitudes
The magnitudes $|\vec{a}|$ and $|\vec{b}|$ represent the lengths of the vectors. By definition, the magnitude of any vector is a non-negative real number: $$|\vec{a}| \ge 0 \quad \text{and} \quad |\vec{b}| \ge 0$$ Given the condition $\vec{a} \cdot \vec{b} \ge 0$, we substitute the definition into the inequality:
$$|\vec{a}| |\vec{b}| \cos \theta \ge 0$$Step 3: Determining the Sign of the Cosine Function
Since the product of magnitudes $|\vec{a}| |\vec{b}|$ is always non-negative, the sign of the dot product depends solely on the value of $\cos \theta$. To satisfy $\vec{a} \cdot \vec{b} \ge 0$, we must have: $$\cos \theta \ge 0$$ In the domain $0 \le \theta \le \pi$, the cosine function is non-negative within the interval $[0, \frac{\pi}{2}]$.
Step 4: Graphical Representation
Step 5: Logical Synthesis
- If $\theta = 0$, $\cos \theta = 1 > 0 \implies \vec{a} \cdot \vec{b} > 0$ (Vectors are parallel/codirectional).
- If $0 < \theta < \frac{\pi}{2}$, $\cos \theta > 0 \implies \vec{a} \cdot \vec{b} > 0$.
- If $\theta = \frac{\pi}{2}$, $\cos \theta = 0 \implies \vec{a} \cdot \vec{b} = 0$ (Vectors are orthogonal).
Final Solution: $0 \le \theta \le \frac{\pi}{2}$
More Questions from Class 12 Mathematics Vector Algebra Miscellaneous Exercise on Chapter 10
- Q1: Write down a unit vector in XY-plane, making an angle of $30^{\circ}$ with the positive direction of x-axis.
- Q2: Find the scalar components and magnitude of the vector joining the points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$.
- Q3: A girl walks 4 km towards west, then she walks 3 km in a direction $30^{\circ}$ east of north and stops. Determine the girl’s displacement from her initial point of departure.
- Q4: If $\vec{a} = \vec{b} + \vec{c}$, then is it true that $|\vec{a}| = |\vec{b}| + |\vec{c}|$? Justify your answer.
- Q5: Find the value of $x$ for which $x(\hat{i} + \hat{j} + \hat{k})$ is a unit vector.
- Q6: Find a vector of magnitude 5 units, and parallel to the resultant of the vectors $\vec{a} = 2\hat{i} + 3\hat{j} - \hat{k}$ and $\vec{b} = \hat{i} - 2\hat{j} + \hat{k}$.
- Q7: If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = 2\hat{i} - \hat{j} + 3\hat{k}$ and $\vec{c} = \hat{i} - 2\hat{j} + \hat{k}$, find a unit vector parallel to the vector $2\vec{a} - \vec{b} + 3\vec{c}$.
- Q8: Show that the points A (1, – 2, – 8), B(5, 0, –2) and C(11, 3, 7) are collinear, and find the ratio in which B divides AC.
- Q9: Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are $2\vec{a} + \vec{b}$ and $\vec{a} - 3\vec{b}$ externally in the ratio 1 : 2. Also, show that P is the mid point of the line segment RQ.
- Q10: The two adjacent sides of a parallelogram are $2\hat{i} - 4\hat{j} + 5\hat{k}$ and $\hat{i} - 2\hat{j} - 3\hat{k}$. Find the unit vector parallel to its diagonal. Also, find its area.
- Q11: Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are $\pm \left( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right)$.
- Q12: Let $\vec{a} = \hat{i} + 4\hat{j} + 2\hat{k}$, $\vec{b} = 3\hat{i} - 2\hat{j} + 7\hat{k}$ and $\vec{c} = 2\hat{i} - \hat{j} + 4\hat{k}$. Find a vector $\vec{d}$ which is perpendicular to both $\vec{a}$ and $\vec{b}$, and $\vec{c} \cdot \vec{d} = 15$.
- Q13: The scalar product of the vector $\hat{i} + \hat{j} + \hat{k}$ with a unit vector along the sum of vectors $2\hat{i} + 4\hat{j} - 5\hat{k}$ and $\lambda\hat{i} + 2\hat{j} + 3\hat{k}$ is equal to one. Find the value of $\lambda$.
- Q14: If $\vec{a}, \vec{b}, \vec{c}$ are mutually perpendicular vectors of equal magnitudes, show that the vector $\vec{a} + \vec{b} + \vec{c}$ is equally inclined to $\vec{a}, \vec{b}$ and $\vec{c}$.
- Q15: Prove that $(\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b}) = |\vec{a}|^2 + |\vec{b}|^2$, if and only if $\vec{a}, \vec{b}$ are perpendicular, given $\vec{a} \neq \vec{0}, \vec{b} \neq \vec{0}$.
- Q17: Let $\vec{a}$ and $\vec{b}$ be two unit vectors and $\theta$ is the angle between them. Then $\vec{a} + \vec{b}$ is a unit vector if
- Q18: The value of $\hat{i} \cdot (\hat{j} \times \hat{k}) + \hat{j} \cdot (\hat{i} \times \hat{k}) + \hat{k} \cdot (\hat{i} \times \hat{j})$ is
- Q19: If $\theta$ is the angle between any two vectors $\vec{a}$ and $\vec{b}$, then $|\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}|$ when $\theta$ is equal to
CBSE Solutions for Class 12 Mathematics Vector Algebra
Chapters in CBSE - Class 12 Mathematics
Top Tutors who teach Vector Algebra
Top Tutor
Sukhlia, Indore
4 yrs of
Exp
2 students
109 hrs of classes completed
500
per hour
Online Classes
I am pursuing a Ph.D. in Economics from DAVV Indore. I have completed Master of Business Administration. I have also completed a Bachelor of Engineering in Computer Science. I am UGC NET qualified in Economics. and I have more than 4 years of experience in teaching.
Very good. Focuses on conceptual and practical learning. Sir helped me to get better in macroeconomics
Top Tutor
Horamavu, Bangalore
4 yrs of
Exp
7 students
176 hrs of classes completed
400
per hour
Online Classes
Tutor's home
As a technology and data professional, my approach to teaching Class 12 Mathematics bridges the gap between abstract textbook formulas and clear, logical problem-solving execution. I specialize in training students for both the CBSE Board and State Board examinations, ensuring a highly structured approach to scoring maximum marks. My Class 12 Mathematics Instructional Blueprint covers: • Calculus Mastery: Simplifying high-weightage topics like Continuity & Differentiability, Integrals, and Applications of Derivatives (AoD) through step-by-step logical breakdowns. • Vectors & 3D Geometry: Using visual and coordinate tools to make highly abstract geometric concepts easy to grasp and reproduce in exams. • Algebra & Probability: Covering Matrices, Determinants, and Relations & Functions with complete conceptual clarity. Exam-Targeted Strategy: • Board-Specific Focus: Comprehensive coverage of NCERT for CBSE and prescribed text-books for State Boards, focusing on question-solving patterns from the past 10 years of board papers. • Marks-Scoring Techniques: Teaching students exactly how to present their steps in the answer sheet to claim full step-marking. • Regular Assessments: Weekly topic-wise tests and time-bound mock exams to track speed and accuracy. My teaching style is patient, structured, and collaborative. I ensure that concepts are thoroughly understood before moving to advanced problems, creating a supportive online environment where students feel completely comfortable clearing their doubts. Let's build a strong mathematical foundation and secure top scores in the upcoming Board Exams! Contact me to book a demo session.
I had a few demo sessions with her and i found her really approachable, patient and fun to learn with. She made sure that i understood, was patient and understanding enough. I really enjoyed learning from her.
Top Tutor
Rajarajeshwari Nagar, Bangalore
12 yrs of
Exp
51 students
1960 hrs of classes completed
500
per hour
Online Classes
Student's Home
I have been training students for 12th Board examinations for more than a decade now.
He is outstanding as a teacher, knowledgeable and dedicated. He has helped my daughter so much with his knowledge, that the girl who was just running away from accounts is getting more than 90 percent
Top Tutor
Ashok Nagar D' Souza Layout, Bangalore
5 yrs of
Exp
5 students
851 hrs of classes completed
500
per hour
Online Classes
Total Years of Experience: Over [X years] of experience in teaching across various academic levels, focusing on delivering engaging, student-centered education. 2. Educational Levels Taught: Primary, Secondary, and Senior Secondary levels Undergraduate-level classes (if applicable) Coaching for competitive exams (NEET, JEE, UPSC, etc.) 3. Key Responsibilities: Planning and delivering lessons based on prescribed curriculum. Assessing students’ understanding through regular tests, quizzes, and assignments. Using various teaching methodologies including lectures, group activities, demonstrations, and ICT tools. Providing remedial classes for weak students and enrichment activities for high performers. Maintaining discipline, fostering motivation, and ensuring an inclusive classroom environment. 4. Classroom Management: Successfully managed classrooms of 30–60 students. Promoted active participation and collaborative learning. Ensured timely syllabus completion while maintaining quality of instruction. 5. Communication and Interaction: Maintained regular communication with parents regarding student progress. Participated in parent-teacher meetings, academic planning sessions, and faculty training programs. 6. Technology Integration: Used smart boards, PowerPoint presentations, online quizzes, and digital simulations to enhance learning outcomes. Conducted online classes using platforms like Google Meet, Zoom, and Microsoft Teams during the COVID-19 pandemic. 7. Co-curricular and Administrative Roles: Organized and participated in educational tours, science exhibitions, and debate competitions. Assisted in exam invigilation, results compilation, and record keeping. 8. Achievements and Impact: Improved student performance and pass percentage year-on-year. Received positive feedback from students and parents for teaching effectiveness and approachability. Mentored students who went on to excel in competitive exams and higher education
had a great learning experience under the guidance of my History teacher. Their teaching style is clear, engaging, and well-structured. Complex topics like the Nationalism in India, Partition, and World War themes were explained with such depth and simplicity that it made understanding and remembering easy. They don’t just teach dates and events—they bring history to life with stories, critical thinking, and real-world connections. Regular assessments, personalized feedback, and doubt-clearing sessions really helped me improve my writing and analytical skills. Highly recommended for any Class 12th student aiming for high marks in board exams!
Top Tutor
Gharoli Gharoli, Delhi
16 yrs of
Exp
1 student
213 hrs of classes completed
600
per hour
Online Classes
Tutor's home
I bring with me 17 years of diverse experience in the pharmaceutical industry along with academic expertise, combining strong practical knowledge with a passion for teaching. I am dedicated to simplifying complex concepts and delivering them in an engaging manner, ensuring that students not only gain subject knowledge but also develop the skills, confidence, and potential required to achieve academic excellence and become industry-ready professionals.
Good tech and good behavior and good teaching style and full topic clear and all of best teacher and my side to best teacher.
Other Subjects in CBSE - Class 12
Download free CBSE - Class 12 Mathematics Vector Algebra Miscellaneous Exercise on Chapter 10 worksheets
Download Now
