Closest point to a line and shortest distance from the origin

Closest point to a line and shortest distance from the origin

Closest Point to a Line

  • Understanding the concept of geometry and vectors is instrumental to solving problems regarding the closest point to a line.
  • The closest point to a line question may require you to represent lines in vector form or parametric form.
  • The line is represented as r = a + λb, where a and b are vectors, r is the position vector of a general point on the line and λ is a scalar parameter.
  • The position vector of the closest point on the line to the origin is obtained by setting r perpendicular to b.
  • Be familiar with the process of finding dot products, which may be required when determining if two vectors are perpendicular.

Shortest Distance from the Origin

  • Understanding how to find the shortest distance from a point to a line plays a crucial role in solving these kind of questions.
  • The formula for the shortest distance (d) from the origin to a line given by the vector equation r = a + λb is d = a x b / b , where x denotes the cross product.
  • Cross product and magnitude of a vector are important mathematical tools you need to have for solving these problems.
  • Make sure to understand how to determine cross product of two vectors and calculate their resulting magnitude.

Applications and Problem Solving

  • These concepts are used in a variety of problems including those involving geometry, physics, and engineering applications.
  • Practice various problems involving finding the closest point to a line and shortest distance from the origin to become accustomed to the problem-solving process.
  • Always break down problems into smaller, manageable parts. Begin by identifying the line equation, then find the relevance in terms of vectors and finally applying the concepts of dot product, cross product and magnitude to find solutions.

Tips for Problem Solving

  • Ensure you show your complete working - marks are often awarded for clear and accurate working.
  • Take a systematic and methodical approach when attempting these problems.
  • Make sure to revise the concepts of dot product, cross product and magnitude of vectors as they’re fundamental to solve these problems.
  • Check all calculations and confirm that your answers make sense within the context of the problem. Even small mistakes can be consequential.