Vector from A to B

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.