Perpendicular vectors

Perpendicular Vectors Revision Content

Basics of Perpendicular Vectors

  • Vectors are quantities that have both a magnitude (length) and a direction.
  • Two vectors are perpendicular if the angle between them is 90°.
  • Perpendicularity in vectors is often associated with the concept of orthogonality.

Dot Product and Perpendicular Vectors

  • The Dot Product of two vectors A = a1i + a2j and B = b1i + b2j is calculated as (a1 * b1) + (a2 * b2).
  • If the dot product of two vectors is zero, this implies the vectors are perpendicular.
  • This is because the dot product of two vectors A and B can also be expressed as A   B cosθ, where θ is the angle between the vectors. If θ=90°, cos(90°) = 0, and therefore the dot product is zero.

Calculating the Dot Product

  • Given two vectors in component form A = [a, b] and B = [x, y], their dot product A.B is calculated by multiplying corresponding components and adding the products: A.B = ax + by.
  • It is crucial to take care with negative signs and understand the component form of vectors to calculate the dot product correctly.

Perpendicularity in Geometry and Physics

  • In geometry, perpendicular vectors can help determine right angles between lines or planes.
  • In physics, perpendicular vectors often arise when decomposing a force or velocity into components, or when considering equilibrium conditions.
  • Being able to identify and work with perpendicular vectors is a crucial skill in the application of vectors to problems in these disciplines.

Perpendicular Unit Vectors

  • A unit vector has a magnitude (length) of 1.
  • If given a vector A, the unit vector in the same direction is A/ A .
  • Two unit vectors are perpendicular if their dot product is zero. This is a result of the fact that the dot product of unit vectors equals the cosine of the angle between them, and cos(90°) = 0.
  • Working with unit vectors can often simplify calculations related to perpendicularity.

Problems Involving Perpendicular Vectors

  • Problems involving perpendicular vectors can come in various forms including finding unknowns in vector equations, determining if vectors are perpendicular, and finding a vector that is perpendicular to two given vectors.
  • Drawing a diagram can often be useful to visualise the vectors and their relationship.
  • Always check your final answer back into the original problem to confirm it does make the vectors perpendicular.