Vector algebra is a fundamental topic in mathematics that deals with vectors, which are quantities that have both magnitude and direction. This tutorial will cover the basics of vector algebra, including vector addition, subtraction, multiplication, and more.

Basic Concepts

  • Magnitude: The magnitude of a vector is its length or size.
  • Direction: The direction of a vector is the line it points along.
  • Unit Vector: A unit vector is a vector with a magnitude of 1.

Vector Addition

Vector addition is the process of combining two or more vectors to find their resultant vector. The resultant vector is the vector that represents the sum of the individual vectors.

  • Head-to-Tail Method: This method involves placing the tail of the second vector at the head of the first vector and drawing a line from the tail of the first vector to the head of the second vector. The resultant vector is the vector that starts at the tail of the first vector and ends at the head of the second vector.

Vector Subtraction

Vector subtraction is the process of finding the difference between two vectors. It is similar to vector addition, but instead of adding the vectors, you subtract the second vector from the first.

  • Head-to-Tail Method: This method involves placing the tail of the second vector at the head of the first vector and drawing a line from the tail of the first vector to the head of the second vector. The resultant vector is the vector that starts at the tail of the first vector and ends at the head of the second vector.

Vector Multiplication

Vector multiplication involves multiplying a vector by a scalar (a real number) or by another vector.

  • Scalar Multiplication: This involves multiplying each component of the vector by the scalar.
  • Dot Product: The dot product of two vectors is a scalar quantity that represents the magnitude of the projection of one vector onto the other.
  • Cross Product: The cross product of two vectors is a vector quantity that is perpendicular to both of the original vectors.

For more information on vector algebra, you can check out our Vector Calculus Tutorial.

Vector Diagram