Detailed Notes from Toppers: Introduction to Vectors

1. Definition of a vector

• Definition: A vector is a geometric object that has both magnitude (or length) and direction. It is represented by an arrow, with the length of the arrow representing the magnitude of the vector and the direction of the arrow representing the direction of the vector.

• Examples from NCERT:

  • Displacement: A change in position of an object from one point to another
  • Velocity: Rate of change of displacement with respect to time.
  • Acceleration: Rate of change of velocity with respect to time.
  • Force: An interaction that can change the motion of an object.

2. Vector algebra

• Vector addition: The sum of two vectors is a vector that has the same direction as the first vector and a magnitude that is the sum of the magnitudes of the two vectors.

• Vector subtraction: The difference between two vectors is a vector that has the same direction as the first vector and a magnitude that is the difference of the magnitudes of the two vectors.

• Scalar multiplication: The product of a scalar and a vector is a vector that has the same direction as the vector and a magnitude that is the product of the scalar and the magnitude of the vector.

• Vector multiplication: The cross product of two vectors is a vector that is perpendicular to both vectors and has a magnitude that is the product of the magnitudes of the two vectors and the sine of the angle between them. The dot product of two vectors is a scalar quantity that is the product of the magnitudes of the two vectors and the cosine of the angle between them.

• Examples from NCERT:

  • Addition and Subtraction of vectors : Displacement of a particle moving with a non uniform velocity.
  • Scalar Multiplication of vectors: Work done by a constant force on a particle.
  • Vector (cross) product : Angular Velocity, Angular momentum, Torque.
  • Vector(Dot) product : Projection of one vector on another.

3. Properties of vectors

• Commutativity: The order of vector addition does not matter, i.e. a + b = b + a.

• Associativity: The grouping of vectors in a sum does not matter, i.e. (a + b) + c = a + (b + c).

• Distributivity: The multiplication of a vector by a scalar can be distributed over vector addition, i.e. k(a + b) = ka + kb.

• Magnitude and direction: The magnitude of a vector is a non-negative real number, and the direction of a vector is a unit vector that points in the same direction as the vector.

• Examples from NCERT:

  • Magnitude of a sum of unit vectors will always be less than or equal to 2.
  • Sum of two perpendicular unit vectors is a unit vector.
  • Magnitude of the vector product of two parallel vectors is zero.

4. Unit vectors

• Definition: A unit vector is a vector that has a magnitude of 1.

• Finding a unit vector: To find the unit vector along a given vector, divide the vector by its magnitude.

• Examples from NCERT:

  • Representation of a unit vector along X-axis is $$ \hat{i} = \frac { \vec{A}}{| \vec{A}|} $$ where $$\vec{A} $$ is any non zero vector along x-axis.

5. Position vectors

• Definition: A position vector is a vector that represents the position of a point in space relative to a fixed origin.

• Operations on position vectors: Position vectors can be added, subtracted, and multiplied by scalars to find the positions of other points in space.

• Examples from NCERT:

  • Position vector of the point P with respect to origin o as $$\overrightarrow{OP}=x \hat{i} + y \hat{j} + z \hat{k}$$ where x,y and z are coordinates of P.

6. Collinear and coplanar vectors

• Collinear vectors: Two vectors are collinear if they lie on the same line or if one vector is a multiple of the other vector.

• Coplanar vectors: Three or more vectors are coplanar if they lie in the same plane or if they can be made coplanar by adding or subtracting them.

• Examples from NCERT:

  • If vectors a, b and c are parallel then a X ( b X c) = 0.
  • If vectors a, b and c are coplanar then $$ a.( b X c) = 0 $$

7. Applications of vectors

• Physics: Vectors are used in physics to describe concepts such as force, velocity, acceleration, momentum, and energy.

• Engineering: Vectors are used in engineering to design and analyze structures, machines, and systems.

• Computer graphics: Vectors are used in computer graphics to create 3D models, animations, and virtual reality environments.

• Examples from NCERT:

  • Projectile Motion.
  • Circular motion.
  • Resolution of forces in Statics.
  • Work done by a Force.

8. Scalar and vector quantities

• Scalar quantities: Scalar quantities are quantities that have only magnitude, such as mass, temperature, and time.

• Vector quantities: Vector quantities are quantities that have both magnitude and direction, such as velocity, acceleration, and force.

• Examples from NCERT:

  • Speed, mass and temperature are Scalar quantities.
  • Velocity, displacement and force are Vector quantities.

9. Resolution of vectors

• Definition: The resolution of a vector is the process of finding the components of the vector along a set of given axes.

• Finding the resultant vector: The resultant vector of a set of vectors is the vector that is equal to the sum of the vectors.

• Examples from NCERT:

  • Resolution of a force into components.
  • Resolution of displacement into components.

10. Applications in physics

• Force: A force is a vector quantity that can change the motion of an object. The magnitude of a force is the amount of force that is being applied, and the direction of a force is the direction in which the force is being applied.

• Velocity: Velocity is a vector quantity that describes the rate at which an object is moving. The magnitude of a velocity is the speed of the object, and the direction of a velocity is the direction in which the object is moving.

• Acceleration: Acceleration is a vector quantity that describes the rate at which an object’s velocity is changing. The magnitude of an acceleration is the amount by which the object’s velocity is changing per unit time, and the direction of an acceleration is the direction in which the object’s velocity is changing.

• Momentum: Momentum is a vector quantity that describes the amount of motion that an object has. The magnitude of a momentum is the product of the object’s mass and velocity, and the direction of a momentum is the same as the direction of the object’s velocity.

• Examples from NCERT:

  • Motion in a straight line.
  • Projectile motion.
  • Circular motion.

11. Applications in geometry

• Lines: A line can be represented by a vector that is parallel to the line and has a magnitude that is equal to the length of the line.

• Planes: A plane can be represented by a vector that is perpendicular to the plane and has a magnitude that is equal to the distance from the origin to the plane.

• Angles: The angle between two vectors can be found by using the dot product of the vectors.

• Examples from NCERT:

  • Vector equation of a straight line.
  • Vector equation of a plane.
  • Angle between two vectors.