Introduction to Vectors

Vectors and scalars

Some quantities need only a size to describe them — a temperature of 20°C, a mass of 5 kg, a time of 10 seconds. These are called scalars.

Other quantities also need a direction. Saying a plane flies at 800 km/h is incomplete; you need to know which way. Quantities with both a size (magnitude) and a direction are called vectors. Examples include velocity, force and displacement (the straight-line journey from one point to another).

A vector is usually drawn as an arrow: its length shows the magnitude and the way it points shows the direction. Vectors are written in bold (a) or with an arrow (a→).

Column vectors

The neatest way to write a 2D vector is as a column vector:

(x, y) meaning x steps horizontally and y steps vertically.

(Properly this is stacked in a column with x on top of y, but we write it as (x, y) here.)

• A positive x means right; a negative x means left.
• A positive y means up; a negative y means down.

So (3, 2) means 3 right and 2 up, while (−4, −1) means 4 left and 1 down. This is closely related to plotting points on the coordinate plane — but a vector describes a movement, not a fixed location, so the same vector can start anywhere.

Adding vectors

To add two vectors, add their x-parts together and their y-parts together — component by component:

(a, b) + (c, d) = (a + c, b + d)

Geometrically, this is "nose to tail": travel along the first vector, then continue along the second from where you finished. The single vector from the very start to the very end is called the resultant.

Worked example 1: Add (2, 5) + (4, −1).

1. Add the x-parts: 2 + 4 = 6.
2. Add the y-parts: 5 + (−1) = 4.
3. Resultant: (6, 4).

Worked example 2: A boat sails with vector (6, 0) (6 east). A current pushes it (0, −3) (3 south). Find the resultant.

1. x-parts: 6 + 0 = 6. y-parts: 0 + (−3) = −3.
2. Resultant: (6, −3) — the boat actually travels 6 east and 3 south.

Subtracting vectors

To subtract, subtract the components, or equivalently reverse the second vector and add:

(a, b) − (c, d) = (a − c, b − d)

Worked example 3: Work out (7, 3) − (2, 5).

1. x-parts: 7 − 2 = 5.
2. y-parts: 3 − 5 = −2.
3. Result: (5, −2).

Subtraction is how you find the vector from one point to another. The vector from point A to point B is B − A (position of B minus position of A).

Worked example 4: Points A = (1, 2) and B = (4, 7). Find the vector AB→.

1. AB→ = B − A = (4 − 1, 7 − 2) = (3, 5).
2. So to get from A to B you move 3 right and 5 up.

Scalar multiplication

Multiplying a vector by a plain number (a scalar) stretches or shrinks it, keeping the same direction (or reversing it if the scalar is negative). Multiply each component:

k × (x, y) = (kx, ky)

Worked example 5: Work out 3 × (2, −1).

1. Multiply each part: 3 × 2 = 6, 3 × (−1) = −3.
2. Result: (6, −3) — three times as long, same direction.

Worked example 6: Work out −2 × (3, 1).

1. −2 × 3 = −6, −2 × 1 = −2.
2. Result: (−6, −2) — twice as long, but pointing the opposite way.

A negative scalar reverses the direction, which is exactly why subtraction works: subtracting a vector is the same as adding −1 times it.

Magnitude: the length of a vector

The magnitude of a vector is its length, written |a|. For a column vector (x, y), it forms a right-angled triangle with horizontal side x and vertical side y, so by the Pythagorean theorem:

|a| = √(x² + y²)

The direction is dropped — magnitude is just how far.

Worked example 7: Find the magnitude of (3, 4).

1. |a| = √(3² + 4²) = √(9 + 16) = √25 = 5.
2. The vector has length 5.

Worked example 8: Find the magnitude of (5, −12).

1. |a| = √(5² + (−12)²) = √(25 + 144) = √169 = 13.
2. The vector has length 13. (The square removes the negative sign, so direction does not affect length.)

When the answer is not a perfect square, leave it as a surd — see surds and irrational numbers.

Combining operations

Real vector questions usually mix the operations. Work through them carefully, one step at a time.

Worked example 9: With a = (3, 1) and b = (1, −2), find 2a + 3b.

1. Scale each first: 2a = 2(3, 1) = (6, 2) and 3b = 3(1, −2) = (3, −6).
2. Add the results component by component: (6 + 3, 2 + (−6)) = (9, −4).
3. So 2a + 3b = (9, −4).

Worked example 10: A parallelogram has vertices P, Q, R, S. The vector PQ→ = (5, 2) and PS→ = (1, 4). Find the vector PR→ (the diagonal).

1. In a parallelogram, the diagonal from P is the sum of the two side vectors leaving P.
2. PR→ = PQ→ + PS→ = (5 + 1, 2 + 4) = (6, 6).
3. The diagonal vector is (6, 6), with magnitude √(6² + 6²) = √72 = 6√2 ≈ 8.49.

This "nose-to-tail" addition is exactly how vectors model the combined effect of two movements or forces.

A reference table

Operation	Rule	Example
Add	(a, b) + (c, d) = (a+c, b+d)	(1, 2) + (3, 4) = (4, 6)
Subtract	(a, b) − (c, d) = (a−c, b−d)	(5, 6) − (2, 1) = (3, 5)
Scalar ×	k(x, y) = (kx, ky)	4(1, −2) = (4, −8)
Magnitude	√(x² + y²)	\	(6, 8)\	= √100 = 10

Where vectors are used

Vectors are the language of physics and engineering. Forces, velocities and accelerations are all vectors, and adding them shows the combined effect — for instance, how wind and engine thrust together set an aircraft's true path. Video games and computer graphics use vectors to move characters and aim projectiles. GPS and navigation track displacement as vectors. Anywhere both how much and which way matter together, vectors are the right tool.

Why direction matters: Two forces of 10 newtons can cancel out (if opposite) or add to 20 (if aligned). A plain number could never capture that — only a vector, carrying direction as well as size, gets the answer right.

Practice activity

Let a = (4, 1) and b = (2, −3).

1. Work out a + b.
2. Work out a − b.
3. Work out 3a.
4. Find the magnitude of b.
5. Points P = (2, 1) and Q = (5, 5). Find the vector PQ→ and its magnitude.

Answers:

1. (4 + 2, 1 + (−3)) = (6, −2).
2. (4 − 2, 1 − (−3)) = (2, 4).
3. (3 × 4, 3 × 1) = (12, 3).
4. √(2² + (−3)²) = √(4 + 9) = √13 ≈ 3.61.
5. PQ→ = (5 − 2, 5 − 1) = (3, 4), magnitude √(9 + 16) = √25 = 5.

Summary

A vector has both magnitude and direction; a scalar has size only. Written as a column vector (x, y), it describes a movement of x across and y up. Add or subtract vectors component by component, multiply by a scalar to stretch, shrink or reverse them, and find the magnitude (length) with √(x² + y²) from Pythagoras. The vector from A to B is B − A. Because they carry direction as well as size, vectors are essential across physics, navigation and computer graphics.