AAHL 3.15.2 Distance between a point and a line

In this lesson, the process of finding the shortest distance between a point and a line is explored. The method involves drawing a diagram to visualize the situation, identifying the perpendicular distance, and using the dot product of vectors to establish the relationship between the point and the line.

Key steps include:

• Defining the position vectors of the point and the line.
• Setting up the equation using the dot product, where the dot product of the vector from the point to the line and the direction vector of the line must equal zero: $\text{PA} \cdot \text{direction vector} = 0$
• Solving for the parameter $\lambda$ to find the coordinates of the point on the line that is closest to the given point.

Finally, the of the vector PA is calculated to determine the shortest distance, which is expressed as: This process emphasizes the complexity of what may initially seem like a straightforward problem.