Projection of Vectors

Duration: 8 min

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Module outline

  1. About Course: About the Course
  2. General Aptitude & Reasoning: Number System, Percentage, Ratio and Proportion (Ratios), Average, Profit and Loss, Simple Interest and Compound Interest, Time and Work, Speed Time and Distance, Mixture and Alligation, Data Interpretation, Algebra, Probability, Permutation and Combination, Venn Diagram, HCF LCM, Divisibility Rules, Age Problems, Pipe and Cistern, Partnership, Linear Algebra, Sequence and Series (Series), Statistics, Powers and Exponents (Surds and Indices), Arithmetic, Mensuration and Geometry, Calculus, Trigonometry, Vectors, Speed Mathematics and Basic Mathematics, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Analogy, Series (Number and Letter Series) (Numerical Relations and Reasoning), Puzzle Solving (Puzzles), Syllogisms, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Calendar, Seating Arrangements, Coding Decoding, Blood Relations, Directions (Direction Test), Clock, Analytical Reasoning (Counting Figures Reasoning), Ranking, Order and Sequence, Logical Sequence Of Words, Inequality, Data Sufficiency, Cubes & Dices
  3. Computer Networks & Cybersecurity: Introduction to CN, Data Communication, DLL: Access Control, DLL: Flow Control, DLL: Error Control, DLL: Framing, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Net Layer:Routing Protocol, Transport Layer Services, TL: Congestion & UDP, Application Layer, Hardware Basics, Network Security
  4. Programming, Data Structures & Algorithms: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling, OOPS with C++, Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing, Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos, Advanced Algorithms
  5. AI, Machine Learning, Big Data & Cloud: Emerging Tech Trends, Artificial Intelligence, Big Data Systems, Approaches to AI, Search Algorithms, Knowledge Representation, Cloud Computing
  6. Digital Logic & Computer Organization: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Basics of COA, Floating Point Rep, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Register Transfer and Microoperations, Control Unit Design, Programming the Basic Computer
  7. TOC & Compiler: Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Grammar, Regular Language Properties, Moore & Mealy Machines, Pushdown Automata & CFG, Turing Machines, Complexity Theory, Intro to Compilers, Lexical Analysis, Grammar & CFG, Syntax Analysis: Top-Down, Syntax Analysis: Bottom-Up, Semantic Analysis & SDT, Intermediate Code Gen, Code Optimization, Run Time Environment
  8. Discrete Mathematics: Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic
  9. Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management, Unix OS, Linux OS, Distributed Systems
  10. Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Testing, Software Maintenance and Configuration Management, Object Oriented Technology, HTML Basics, JavaScript-Basics, Java, XML
  11. Database Management System: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, File Organization & Indexing, Relational Algebra, SQL, Relational Calculus, Transaction Management, Concurrency Control, Database Recovery, Data Warehousing & Mining, Enhanced Data Models, NoSQL
  12. Other Recurring CS Topics: Basics of Computer Graphics, 2-D Geometrical Transforms and Viewing, Multimedia
  13. Live Classes: ISRO Live classes
  14. Full Mock Test:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This video is a mathematics lecture on vector projection, presented on a digital blackboard. The instructor first defines the scalar projection of vector a on vector b using the formula (a · b) / |b|, which is derived from the dot product and the cosine of the angle between the vectors. The concept is illustrated with a diagram showing two vectors, a and b, and the angle θ between them. The lecture explains that the projection is positive when the angle is acute (θ < 90°), zero when the angle is 90°, and negative when the angle is obtuse (θ > 90°). The video then transitions to a worked example, where the scalar projection of the vector 3i - j - 2k onto the vector i + 2j - 3k is calculated. The instructor applies the formula, computes the dot product (3*1 + (-1)*2 + (-2)*(-3) = 7), and the magnitude of the second vector (√(1² + 2² + (-3)²) = √14), arriving at the final answer of 7/√14.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video begins with a title 'Projection' on a black background. The instructor introduces the concept of the scalar projection of vector a on vector b. A diagram shows two vectors, a and b, with an angle θ between them. The formula for the projection is written as (a · b) / |b|. The instructor explains that this is derived from the dot product formula a · b = |a||b|cosθ. The projection is also described as the component of vector a in the direction of vector b. The video shows the formula for the projection of b on a as (b · a) / |a|, and notes that the projection is positive if θ < 90°, zero if θ = 90°, and negative if θ > 90°. The instructor uses a laser pointer to highlight the formula and the diagram.

  2. 2:00 – 5:00 02:00-05:00

    The instructor continues to explain the projection formula, emphasizing that the projection of a on b is a scalar quantity. The formula is written as 'proj of a on b = (a · b) / |b|'. The instructor explains that this is the length of the projection of vector a onto vector b. The video shows the formula for the projection of b on a as (b · a) / |a|, which is the same as (a · b) / |a|. The instructor explains that the projection is positive when the angle between the vectors is acute, zero when the angle is 90 degrees, and negative when the angle is obtuse. The instructor uses a laser pointer to highlight the different parts of the formula and the diagram.

  3. 5:00 – 7:38 05:00-07:38

    The video transitions to a worked example. The problem is stated: 'The scalar projection of the vector 3i - j - 2k on the vector i + 2j - 3k is:'. The instructor identifies vector a as 3i - j - 2k and vector b as i + 2j - 3k. The formula for the scalar projection is written as (a · b) / |b|. The dot product a · b is calculated as (3*1) + (-1*2) + (-2*-3) = 3 - 2 + 6 = 7. The magnitude of vector b is calculated as √(1² + 2² + (-3)²) = √(1 + 4 + 9) = √14. The final answer is (7) / (√14). The instructor then simplifies this to 7/√14, which is one of the multiple-choice options provided on the screen.

The video provides a clear and structured lesson on vector projection. It begins with a theoretical explanation of the scalar projection, using a diagram and the dot product formula to derive the key equation. The instructor then demonstrates the application of this formula through a concrete example, walking through each step of the calculation. The progression from theory to practice effectively reinforces the concept, making it accessible for students. The use of a digital blackboard and a laser pointer helps to focus the viewer's attention on the relevant parts of the equations and diagrams.

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