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.