Vector cross product

Determine orientation with the cross product.

Vector cross product

The cross product determines the orientation of three points: left turn (counterclockwise), right turn (clockwise) or colinear. It is the foundation of geometric algorithms.

For points A, B, C the cross product is (B-A) × (C-A). In two dimensions this yields a scalar via (bx-ax)·(cy-ay) - (by-ay)·(cx-ax).

A positive result means left turn, negative means right turn, zero means colinear.

Orientation formula

To compute the orientation of three points we derive two edge vectors: AB = (bx-ax, by-ay) and AC = (cx-ax, cy-ay).

The cross product AB × AC is the determinant of these two vectors. The result is the signed area of the parallelogram.

This computation is O(1) and is used constantly in more complex algorithms.

Exercise cross_product function

Write cross_product(ax, ay, bx, by, cx, cy). Return the cross product of three points. Print 12 with print(cross_product(0, 0, 4, 0, 2, 3)).

Show example solution

def cross_product(ax, ay, bx, by, cx, cy):
    # Cross product of three points
    return (bx - ax) * (cy - ay) - (by - ay) * (cx - ax)
print(cross_product(0, 0, 4, 0, 2, 3))

Test yourself

What does a negative cross product mean?
```
cp = (bx-ax)*(cy-ay) - (by-ay)*(cx-ax)
```
- Left turn (counterclockwise)
- Colinear
- Zero
- Right turn (clockwise)
Show answer

Right turn (clockwise) — A negative cross product means right turn (clockwise).

cross product computation

cp = (bx - ax) * (cy - ay) - (____ - ay) * (cx - ax)

Show answer

cp = (bx - ax) * (cy - ay) - (by - ay) * (cx - ax)

Orientation concepts

positive

Left turn
negative

Right turn
zero

Colinear
determinant

Signed area

Next lesson: Convex hull

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