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Collinearity of two vectors and alignment of points

Level 5 · Ages 15–16

Quick recall· Let's wake up what you already know before we start.

Review: A(1, 2) and B(4, 3). What are the coordinates of vector AB?

The whole lesson

Read it again any time, including while you work through the exercises.

You already know how to read a vector as a displacement: move by x horizontally and y vertically. Now look at two arrows on a grid: u(2, 3) and v(4, 6). The second one moves exactly twice as far in the same direction: v = 2u. When one vector is a number times the other, we say they are collinear: their arrows are parallel, they follow the same direction.

Spotting that number by eye is not always easy, especially with signs or decimals. Luckily, there is a foolproof test: vectors u(x, y) and v(x', y') are collinear if and only if xy' − yx' = 0. This number is called the determinant of u and v. It crosses the coordinates in an X pattern: the first coordinate of one multiplies the second coordinate of the other, then you subtract.

What is it for? First, spotting parallel lines: two lines directed by collinear vectors are parallel. Then, and this is the best part, proving that three points A, B, C are aligned: it is enough to show that vectors AB and AC are collinear. Since they both start from A, if they point in the same direction, the three points must lie on the same line.

Picture the method on the grid: I compute the coordinates of AB and AC (arrival minus start), I cross them with the determinant, and I conclude. Zero: aligned. Not zero: not aligned. One single computation replaces a whole drawing, that is the power of coordinate geometry. Your turn!

The worked example, step by step

Are vectors u(2, 3) and v(4, 6) collinear?

  1. xy' − yx' = 2 × 6 − 3 × 4

    I apply the collinearity test: I cross the coordinates in an X pattern, first coordinate times the other's second, then subtract.

  2. 12 − 12 = 0

    The determinant is 0: the vectors are collinear. I can even spot the coefficient: v = 2u, arrow v is twice as long in the same direction.

Result: 2 × 6 − 3 × 4 = 0: u and v are collinear (v = 2u).

Are the points A(1, 2), B(3, 5) and C(7, 11) aligned?

  1. AB(2 ; 3)

    I compute vector AB: (3 − 1, 5 − 2) = (2, 3).

  2. AC(6 ; 9)

    Then vector AC: (7 − 1, 11 − 2) = (6, 9). Both vectors start from the same point A.

  3. 2 × 9 − 3 × 6 = 18 − 18 = 0

    I test collinearity with the determinant: it is 0, so AB and AC are collinear.

  4. A, B, C alignés

    AB and AC are collinear and both start from A: the three points therefore lie on the same line. Indeed AC = 3AB.

Result: The determinant of AB and AC is 0: A, B and C are aligned.