- Find the coordinates and the length of a vector
- Add, subtract and scale vectors, both with coordinates and geometrically
- Use the dot product to find angles and test for perpendicularity
A boat crosses the Kura river: it moves at 4 m/s straight across, while the current carries it downstream at 3 m/s. What is the boat's real speed? Not 4 + 3 = 7 m/s, but 5 m/s! A number alone is not enough to describe velocity; its direction matters too. Quantities that have both a size and a direction, such as velocity, force and displacement, are described by vectors; mass and temperature are ordinary numbers (scalars).
Vectors and their coordinates
A directed line segment: it has a start, an end, a length (magnitude) and a direction. The vector AB⃗ starts at A and ends at B; a vector can also be named by one letter: a⃗. Vectors with the same length and the same direction are equal; vectors lying on the same line or on parallel lines are called collinear (parallel).
- A(x₁, y₁)the start of the vector
- B(x₂, y₂)the end of the vector
To find the coordinates of a vector, subtract the coordinates of the start from those of the end (“end minus start”). The rule works in space too.
- x, ythe coordinates of a⃗
- |a⃗|the length (magnitude) of the vector
This is the Pythagorean theorem. In space a vector has three coordinates and |a⃗| = √(x² + y² + z²): for example, the vector (2, 3, 6) has length √(4 + 9 + 36) = 7.
Given A(1, −2) and B(4, 2), find the coordinates and the length of AB⃗.
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|AB⃗| = √(3² + 4²) = √25 = 5.
This is also the distance between the points A and B.
Operations with vectors
- a⃗ = (x₁, y₁), b⃗ = (x₂, y₂)the given vectors
- ka number (scalar)
Vectors are added and subtracted coordinate by coordinate; multiplying by a number multiplies every coordinate by it.
Geometrically, two vectors are added with the triangle rule (place the start of b⃗ at the end of a⃗) or the parallelogram rule (the diagonal drawn from their common start). For k > 0 the vector k · a⃗ points the same way as a⃗, for k < 0 the opposite way, and its length is |k| · |a⃗|.
- x₁, y₁, x₂, y₂the coordinates of a⃗ and b⃗
Collinear vectors have proportional coordinates: b⃗ = k · a⃗. The second form also works when a coordinate is zero.
a⃗ = (2, −1), b⃗ = (−3, 4).
a) Find 2a⃗ − b⃗ and its length.
b) For which m is c⃗ = (m, 3) collinear with a⃗?
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|2a⃗ − b⃗| = √(49 + 36) = √85 ≈ 9.22.
b) The coordinates must be proportional: m/2 = 3/(−1) ⇒ m = −6.
Check: c⃗ = (−6, 3) = −3 · a⃗, so c⃗ points opposite to a⃗.
The dot product
- φthe angle between the vectors, 0° ≤ φ ≤ 180°
- |a⃗|, |b⃗|the lengths of the vectors
The dot product is not a vector but a number (a scalar), which is why it is also called the scalar product. a⃗ · a⃗ = |a⃗|².
- x₁, y₁, x₂, y₂the coordinates of a⃗ and b⃗
In coordinates, the dot product is the sum of the products of matching coordinates. In space, add z₁z₂.
- a⃗, b⃗non-zero vectors
The sign of the dot product tells you the angle: > 0 acute, = 0 right angle (perpendicular), < 0 obtuse.
Find the angle between a⃗ = (1, 2) and b⃗ = (3, 1).
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|a⃗| = √(1 + 4) = √5, |b⃗| = √(9 + 1) = √10.
cos φ = 5 / (√5 · √10) = 5 / √50 = 5 / (5√2) = √2/2 ⇒ φ = 45°.
a) For which k are a⃗ = (k, 2) and b⃗ = (3, −6) perpendicular?
b) A force of 50 N makes an angle of 60° with a displacement of 10 m. Find the work done by the force.
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b) Work is the dot product of force and displacement: W = F⃗ · s⃗ = F · s · cos φ = 50 · 10 · cos 60° = 50 · 10 · 0.5 = 250 J.
Key points
- AB⃗ = (x₂ − x₁, y₂ − y₁), “end minus start”; |a⃗| = √(x² + y²).
- Vectors are added and subtracted coordinate by coordinate; k · a⃗ multiplies each coordinate by k.
- Chain rule: AB⃗ + BC⃗ = AC⃗.
- a⃗ · b⃗ = |a⃗| · |b⃗| · cos φ = x₁x₂ + y₁y₂.
- a⃗ ⊥ b⃗ ⇔ x₁x₂ + y₁y₂ = 0; a⃗ ∥ b⃗ ⇔ the coordinates are proportional.
Check yourself
10 questions. Every correct answer earns XP.