Vectors

Fundamental Concepts of Vectors

  • Vectors: An entity with both magnitude and direction. Unlike scalars, which only have magnitude.
  • Position Vector : A vector describing a position relative to an origin.
  • Magnitude of a Vector: Also known as the length or absolute value of a vector, calculated using Pythagoras theorem for 2D, extended for 3D.

Vectors and Coordinate Geometry

  • Unit Vector : A vector with a magnitude of 1. Represented using a hat (^), such as i, j , k for the x, y, and z direction respectively.
  • Direction Cosines: Cosines of the angles between the vector and the coordinate axes.
  • Parallel and Perpendicular Vectors: Two vectors are parallel if one is a scalar multiple of the other. Two vectors are perpendicular if their dot product equals zero.

Operations with Vectors

  • Vector Addition & Subtraction: The process of combining or subtracting magnitudes and directions.
  • Scalar Multiplication: The process of scaling a vector by multiplying its magnitude by a scalar.
  • Dot Product: The multiplication of the magnitude and cosine of the angle between two vectors.
  • Cross Product: The product of the magnitudes and sine of the angle, results in a new vector perpendicular to the plane formed by the two vectors.

Vectors and Geometry

  • Definition of a Line: A line can be defined by a point on the line and a direction vector parallel to the line.
  • Intersection of Lines: Two lines intersect if there exists a common solution to the parametric equations of the two lines.
  • Angle between Two Lines: Defined as the acute angle that one line makes with another.
  • Definition of a Plane: A plane can be defined using a point in the plane and a normal vector.
  • Intersection of a Line and a Plane: A line intersects a plane if there exists a common solution to the parametric equation of the line and the Cartesian equation of the plane.

Applications of Vectors

  • Displacement: A vector that specifies the change in position of a point relative to a reference.
  • Velocity: A vector expression of the displacement that an object experiences per unit time.
  • Force: A vector that represents the push or pull experienced by an object resulting from its interaction with another object.
  • Work Done: Scalar quantity obtained from the dot product of force and displacement vector.