Vector from A to B

GCSE Vectors vectors position vector
Practice this formula All formulas
\( \overrightarrow{AB}=\begin{pmatrix}x_2-x_1\\ y_2-y_1\end{pmatrix} \)

Statement

If you have two points in the plane, \( A(x_1,y_1) \) and \( B(x_2,y_2) \), then the vector from A to B is given by subtracting the coordinates of A from those of B:

\[ \overrightarrow{AB} = \begin{pmatrix}x_2 - x_1 \\ y_2 - y_1\end{pmatrix} \]

This tells us the direction and displacement needed to move from A to B.

Why it’s true

  • A vector represents a displacement, or movement, from one point to another.
  • To move from A to B, you travel \(x_2-x_1\) units horizontally and \(y_2-y_1\) units vertically.
  • This works in all quadrants and for positive or negative differences.

Recipe (how to use it)

  1. Identify coordinates of A and B.
  2. Subtract: x-coordinate of A from x-coordinate of B; y-coordinate of A from y-coordinate of B.
  3. Write the result as a column vector.

Spotting it

Look for phrasing like “vector from A to B” or “displacement from one point to another”.

Common pairings

  • Often followed by finding magnitude of the vector (distance).
  • Also used before forming unit vectors or equations of lines.

Mini examples

  1. Given: A(1,2), B(4,6). Find: \(\overrightarrow{AB}\). Answer: (3,4).
  2. Given: A(-2,5), B(1,1). Find: \(\overrightarrow{AB}\). Answer: (3,-4).

Pitfalls

  • Subtracting in the wrong order (remember: B minus A).
  • Forgetting negative signs when coordinates are negative.
  • Confusing vector with midpoint or distance.

Exam strategy

  • Label points clearly before subtracting.
  • Check direction: \(\overrightarrow{AB}\) is opposite to \(\overrightarrow{BA}\).
  • Simplify your vector if it has common factors, especially in multi-step problems.

Summary

The vector from A to B is simply the difference in their coordinates. Subtract A from B to find the displacement in x and y. This formula is fundamental in vector geometry and underpins distances, directions, and line equations.