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Chapter 4 – Vectors And Three Dimensional Geometry

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:


You might have come across Vectors And Three Dimensional Geometry while solving your maths problems in your geometry class. But have you faced difficulty in solving them easily? If yes, then we are here to aid you in understanding the core concept behind them. In this article, we will discuss everything about Vectors And Three Dimensional Geometry. We hope that this article will help clear your doubts. This topic is very important to understand different types of shapes and figures in geometry.

It depends on a two-dimensional plane consisting of the x-axis and the y-axis. Opposite to one another, the axes partition the plane into four areas. Each part is known as a quadrant. There are four quadrants. Namely:-

Quadrant I

Quadrant II

Quadrant III

Quadrant IV

The two-dimensional plane where the 

  • the x-axis is the horizontal axis
  • the y-axis is the vertical axis 

A point in the plane is characterized as an arranged pair (x,y), with the end goal that x is controlled by its flat separation from the beginning, and y is dictated by its vertical separation from the source.

We can find Distance Between 2 Points by using the distance formula. The formula is d=√((x_2-x_1)²+(y_2-y_1)²).  Here, the Pythagorean theorem is applied. The best way to learn this formula is to set up a correct triangle and utilize the Pythagorean hypothesis at whatever point they need to discover the distance between two focuses.

As an uncommon instance of the distance equation, assume we need to know the distance of a point (x,y) to the origin. As per the distance recipe, this is √(x−0)2+(y−0)2=√x2+y2. A point (x,y) is a way off r from the origin if and just if √x2+y2=r, or if we square the two sides: x2+y2=r2.

In Maths, a triangle is a three-sided polygon that has three edges and three vertices. The territory of the triangle is a proportion of the space canvassed by the triangle in the two-dimensional plane. At the point when we have vertices of the triangle, and we need to discover the zone of the triangle, we can follow the following steps:-

  • The plot focuses on a chart. 
  • Take the vertices in a counterclockwise direction. In any case, the equation gives a negative worth. 
  • Use the equation A = Hbb/2.

The cosines of a line are the cosines of the points which the line makes with the positive bearings of the facilitated coordinate axes. 

Here it is noted that the direction cosines for any line should be exceptional. Be that as it may, there are limitlessly numerous arrangements of bearing proportions since numbers proportions are only a set of any three numbers relative to the direction cosines.

Parallel lines will be lines in a plane that are consistently a similar distance separated. Parallel lines won’t ever meet. Parallel planes will be planes in the very three-dimensional space that won’t ever meet. Parallel lines can be horizontal, vertical, or diagonal. 

When two straight lines are plotted on the coordinate plane, we can tell if they are parallel from the slope of each line. If the slopes are the same, then the lines are parallel.

The projection here signifies “The portrayal of a figure or strong on a plane as it would look from a specific course”. The projection of line can be of two types:-

  • Not belonging to a line: The projection of line when it is outer to the line.
  • Belonging to a line: The projection of line when it is on the line.

Question 1. What is a three-dimensional vector?

Answer. A three-dimensional vector is a line segment in three-dimensional space running from point A to point B. Each vector has a magnitude and direction.

Question 2. What is a Vector?

Answer.  A vector is a coordinated line section whose length is the extent of the vector and with a bolt showing the heading. The course of the vector is from its tail to its head.

Question 3. What are the examples of parallel lines in real life?

Answer. We can see parallel lines in real life in a notebook, railway tracks, zebra crossing, etc. 

Question 4. What are the main properties of parallel lines?

Answer.  The following are the main properties of parallel lines:-

  • The comparing points are equivalent. 
  • The vertically inverse points are equivalent. 
  • The other inside points are equivalent. 
  • The other outside points are equivalent. 
  • The pair of inside points on a similar side of the cross-over is strengthening.

Question 5. Define the Cartesian vector form.

Answer.  The vector, being the amount of the vectors and, is subsequently. This equation, which communicates j, k, x, y, and z, is known as the Cartesian portrayal of the vector in three-dimension.

Many questions are asked from Vectors And Three Dimensional Geometry in class 12 board exams. Formulas are critical so understand them by practising as many numerical as possible. Topics are easy and can be completed effortlessly. For your professional growth, it is important to have a proper understanding of this topic. You can check the MSVgo app to know more about the topic. The MSVgo philosophy is to enable a core understanding of any concept. MSVgo is a video library that explains concepts with examples or explanatory visualizations or animations. Check out videos on MSVgo to understand the concept, behind Vectors And Three Dimensional Geometry.

High School Physics

  • Alternating Current
  • Atoms
  • Communication Systems
  • Current Electricity
  • Dual nature of Radiation and Matter
  • Electric Charges and Fields
  • Electricity
  • Electromagnetic Induction
  • Electromagnetic Waves
  • Electron Beams and Radioactivity
  • Electrons and Photons
  • Electrostatic Potential and Capacitance
  • Fluid Pressure
  • Force and Acceleration
  • Force And Laws Of Motion
  • Gravitation
  • Internal Energy
  • Kinetic Theory
  • Law of motion
  • Light – Reflection And Refraction
  • Magnetic Effects Of Electric Current
  • Magnetism and Matter
  • Management Of Natural Resources
  • Mechanical properties of Fluids
  • Mechanical properties of Solids
  • Motion
  • Motion in a plane
  • Motion in a straight line
  • Moving Charges and Magnetism
  • Nuclear Energy
  • Nuclei
  • Oscillations
  • Our Environment
  • Paths of Heat
  • Physical world
  • Ray optics and optical instruments
  • Semiconductor Devices
  • Semiconductor Electronics: Materials, Devices and Simple Circuits
  • Simple Machines
  • Sound
  • Sources Of Energy
  • Specific and Latent Heats
  • Spherical Mirrors
  • Static Electricity
  • Systems of Particles and Rotational motion
  • Thermal properties of matter
  • Thermodynamics
  • Units and Measurement
  • Vectors, Scalar Quantities and Elementary Calculus
  • Wave Optics
  • Waves
  • Work, Power and Energy

High School Chemistry

  • Acids, Bases and Salts
  • Alcohols, Phenols and Ethers
  • Aldehydes, Ketones and Carboxylic Acids
  • Aliphatic and Aromatic Hydrocarbons
  • Alkyl and Aryl Halides
  • Amines
  • Analytical Chemistry 
  • Atomic Structure
  • Atoms And Molecules
  • Basic concepts of Chemistry
  • Biomolecules
  • Carbon And Its Compounds
  • Carboxylic acids and Acid Derivatives
  • Chemical Bonding and Molecular Structures
  • Chemical Energetics
  • Chemical Equilibria
  • Chemical Kinetics
  • Chemical Reactions And Equations
  • Chemical Reactions and Their Mechanisms
  • Chemistry in Everyday Life
  • Chemistry of p-Block elements
  • Chemistry of Transition and Inner Transition
  • Classification of Elements
  • Coordination Compounds
  • Cyanide, Isocyanide, Nitro compounds and Amines
  • Electrochemistry
  • Electrolysis
  • Elements, Compounds and Mixtures
  • Environmental Chemistry
  • Equilibrium
  • Ethers and Carbonyl compounds
  • Haloalkanes and Haloarenes
  • Hydrocarbons
  • Hydrogen
  • Ideal solutions
  • Introduction to Organic Chemistry
  • Ionic equilibria
  • Matter
  • Matter Around Us
  • Matter In Our Surroundings
  • Metallurgy
  • Metals And Non-Metals
  • Mole Concept and Stoichiometry
  • Natural Resources
  • Organic Chemistry – Basic Principles
  • Periodic Classification of Elements
  • Physical and Chemical Changes
  • Physical and Chemical Properties of Water
  • Polymers
  • Preparation, Properties and Uses of Compounds
  • Principles and Processes of Isolation of Elements
  • Redox Reactions
  • Relative Molecular Mass and Mole
  • States of Matter
  • Structure Of The Atom
  • Study of Compounds
  • Study of Gas Laws
  • Study of Representative Elements
  • Surface Chemistry
  • The d-block and f-block elements
  • The Gaseous State
  • The p-Block Elements
  • The Periodic Table
  • The s-Block Elements
  • The Solid State
  • Thermodynamics

High School Biology

  • Absorption and Movement of Water in Plants
  • Adolescent Issues
  • Anatomy of Flowering Plants
  • Animal Kingdom
  • Bacteria and Fungi-Friends and Foe
  • Biodiversity and Conservation
  • Biofertilizers
  • Biological Classification
  • Biomedical Engineering
  • Biomolecules
  • Biotechnology and its Applications
  • Biotic Community
  • Body Fluids and Circulation
  • Breathing and Exchange of Gases
  • Cell – Unit of Life
  • Cell Cycle and Cell Division
  • Cell Division and Structure of Chromosomes
  • Cell Reproduction
  • Cellular Respiration
  • Chemical Coordination and Integration
  • Circulation
  • Control And Coordination
  • Crop Improvement
  • Digestion and Absorption
  • Diversity In Living Organisms
  • Ecosystem
  • Environmental Issues
  • Excretory Products and their Elimination
  • Flowering Plants
  • Genes and Chromosomes
  • Health and Diseases
  • Health and Its Significance
  • Heredity And Evolution
  • Heredity and Variation
  • How Do Organisms Reproduce?
  • Human Diseases
  • Human Eye And Colourful World
  • Human Health and Disease
  • Human Population
  • Human Reproduction
  • Hygiene
  • Improvement In Food Resources
  • Integumentary System- Skin
  • Kingdom Fungi
  • Kingdom Monera
  • Kingdom Protista
  • Life Processes
  • Locomotion and Movement
  • Microbes in Human Welfare
  • Mineral Nutrition
  • Molecular Basis of Inheritance
  • Morphology of Flowering Plants
  • Neural Control And Coordination
  • Nutrition in Human Beings
  • Organism and Population
  • Photosynthesis
  • Photosynthesis in Higher Plants
  • Plant Growth and Development
  • Plant Kingdom
  • Pollination and Fertilization
  • Pollution; Sources and its effects
  • Principles of Inheritance and Variation
  • Reproduction and Development in Angiosperms
  • Reproduction in Organisms
  • Reproductive Health
  • Respiration in Human Beings
  • Respiration in Plants
  • Respiratory System
  • Sexual Reproduction in Flowering Plants
  • Strategies for Enhancement in Food Production
  • Structural Organisation in Animals
  • Structural Organisation of the Cell
  • The Endocrine System
  • The Fundamental Unit Of Life
  • The Living World
  • The Nervous System and Sense Organs
  • Tissues
  • Transpiration
  • Transport in Plants

High School Math

  • Algebra – Arithmatic Progressions
  • Algebra – Complex Numbers and Quadratic Equations
  • Algebra – Linear Inequalities
  • Algebra – Pair of Linear Equations in Two Variables
  • Algebra – Polynomials
  • Algebra – Principle of Mathematical Induction
  • Algebra – Quadratic Equations
  • Binomial Theorem
  • Calculus – Applications of Derivatives
  • Calculus – Applications of the Integrals
  • Calculus – Continuity and Differentiability
  • Calculus – Differential Equations
  • Calculus – Integrals
  • Geometry – Area
  • Geometry – Circles
  • Geometry – Conic Sections
  • Geometry – Constructions
  • Geometry – Introduction to Euclid’s Geometry
  • Geometry – Three-dimensional Geometry
  • Geometry – Lines and Angles
  • Geometry – Quadrilaterals
  • Geometry – Straight Lines
  • Geometry – Triangles
  • Linear Programming
  • Matrices and Determinants
  • Mensuration – Areas
  • Mensuration – Surface Areas and Volumes
  • Number Systems
  • Number Systems – Real Numbers
  • Permutations and Combinations
  • Probability
  • Sequence and Series
  • Sets and Functions
  • Statistics 
  • Trignometry – Height and Distance
  • Trignometry – Identities
  • Trignometry – Introduction

Middle School Science

  • Acids, Bases And Salts
  • Air and Its Constituents
  • Basic Biology
  • Body Movements
  • Carbon and Its Compounds
  • Cell – Structure And Functions
  • Changes Around Us
  • Chemical Effects Of Electric Current
  • Chemistry in Your Life
  • Coal And Petroleum
  • Combustion And Flame
  • Components Of Food
  • Conservation Of Plants And Animals
  • Crop Production And Management
  • Electric Current And Its Effects
  • Electricity And Circuits
  • Elements and Compounds
  • Fibre To Fabric
  • Food production and management
  • Force And Pressure
  • Forests: Our Lifeline
  • Friction
  • Fun With Magnets
  • Garbage In, Garbage Out
  • Getting To Know Plants
  • Health and Hygiene
  • Heat
  • Hydrogen
  • Life Processes: Nutrition in Animals and Plants
  • Light, Shadows And Reflections
  • Materials: Metals And Non-Metals
  • Matter and Its States
  • Metals and Non-metals
  • Micro Organisms: Friend And Foe
  • Motion And Measurement Of Distances
  • Motion And Time
  • Nutrition In Animals
  • Nutrition In Plants
  • Organization in Living Things
  • Our Environment
  • Physical And Chemical Changes
  • Pollution and conservation
  • Pollution Of Air And Water
  • Reaching The Age Of Adolescence
  • Reproduction In Animals
  • Reproduction In Plants
  • Respiration In Organisms
  • Rocks and Minerals
  • Separation Of Substances
  • Simple Machines
  • Soil
  • Some Natural Phenomena
  • Sorting Materials Into Groups
  • Sound
  • Stars And The Solar System
  • Structure of Atom
  • Synthetic Fibers And Plastics
  • The Living Organisms And Their Surroundings
  • Transfer of Heat
  • Transformation of Substances
  • Transportation In Animals And Plants
  • Universe
  • Waste-water Story
  • Water: A Precious Resource
  • Weather, Climate And Adaptations Of Animals To Climate
  • Winds, Storms And Cyclones

Middle School Math

  • Addition
  • Area and Its Boundary
  • Boxes and Sketches
  • Data Handling
  • Fun With Numbers
  • Heavy and Light
  • How Many
  • Long And Short
  • Mapping
  • Measurement
  • Money
  • Multiplication and Factors
  • Multiply and Divide
  • Numbers
  • Parts and Wholes
  • Pattern Recognition
  • Patterns
  • Play With Patterns
  • Rupees And Paise
  • Shapes And Angles
  • Shapes And Designs
  • Shapes and Space
  • Similarity
  • Smart Charts
  • Squares
  • Subtraction
  • Tables And Shares
  • Tenths and Hundredths
  • Time
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