What is a vector in maths, and how do you add vectors?

Some quantities need only a number. A mass of 5 kg, a temperature of 18 °C or a speed of 30 km/h are scalars: they have a size but no direction. Other quantities make no sense without a direction. "Walk 3 km" does not tell you where you end up; "walk 3 km east" does. Quantities like this are vectors. Displacement, velocity, force and acceleration are all vectors, which is why physics uses them all the time.

You draw a vector as an arrow. Its length shows the magnitude and the way it points shows the direction. In books a vector is written in bold, a, or underlined by hand, a̲. A vector from point A to point B is written AB with an arrow on top.

In maths at school, most vectors are column vectors. The top number is the move across (right is positive, left is negative) and the bottom number is the move up (up is positive, down is negative). It is written as a column in brackets, with 3 above 4; on this page we write it on one line as (3, 4). It means "3 right, 4 up", the same kind of move as a translation in transformations.

The rules are simple because you work on the two numbers separately:

• Adding: (3, 4) + (1, −2) = (4, 2). On a drawing, put the arrows nose to tail; the answer, the resultant, goes from the start of the first to the end of the last.
• Subtracting: (3, 4) − (1, −2) = (2, 6). Subtracting b is the same as adding −b, the arrow turned round.
• Multiplying by a number (a scalar): 2 × (3, 4) = (6, 8). The arrow becomes twice as long and keeps its direction. A negative number turns it round.
• Magnitude: the length of (x, y) is √(x² + y²), straight from Pythagoras' theorem. The length of (3, 4) is √(9 + 16) = 5.

Two vectors are parallel if one is a number times the other: (2, 3) and (6, 9) are parallel because (6, 9) = 3 × (2, 3). Three points A, B and C are collinear (on one straight line) if AB and BC are parallel and share the point B. Two vectors are perpendicular if their dot product is zero: for (a, b) and (c, d) that is ac + bd = 0.

Vectors also make geometry easier. If O is the origin, the position vector of a point A is OA. With OA = a and OB = b, the vector from A to B is always b − a, and the midpoint of AB has position vector ½(a + b). In 3D a vector simply has three numbers, (x, y, z), and its magnitude is √(x² + y² + z²).

What is the difference between a vector and a scalar?#

A scalar has only a size, like 20 m/s or 5 kg. A vector has a size and a direction, like 20 m/s due north. Speed is a scalar; velocity is the matching vector.

How do you add two vectors?#

Add the top numbers together and the bottom numbers together: (2, 5) + (3, −1) = (5, 4). On a drawing, place the arrows nose to tail and draw the resultant from the very start to the very end.

How do you know if two vectors are parallel?#

One must be a single number times the other, in both rows. (3, −2) and (−6, 4) are parallel because (−6, 4) = −2 × (3, −2); the minus sign means they point opposite ways.

How do you show that two vectors are perpendicular?#

Find the dot product: multiply the top numbers, multiply the bottom numbers and add. If the result is 0, the vectors are at right angles, as with (2, 3) and (3, −2).

How do you show that three points are collinear?#

Show that AB and BC (or AB and AC) are parallel. Because they share a point, they must lie on the same straight line.

What is a vector in physics?#

Any quantity with a direction: displacement, velocity, acceleration, force and momentum. Adding force vectors gives the resultant force, which decides how an object moves.

Does "vector" mean something else in biology or design?#

Yes. In biology a vector is a living thing that carries a disease to people, such as a mosquito carrying malaria. In design, vector graphics store pictures as points, lines and curves described by numbers, so they can be enlarged without going blurry.