Vectors Explained: Addition, Magnitude and Direction

Vectors explained simply: a vector is a quantity with both size and direction, unlike a scalar, which only has size. Velocity, force and displacement are vectors; speed, mass and temperature are scalars. Vectors are usually drawn as arrows: the arrow's length shows the magnitude (size) and the arrow's direction shows, well, the direction. Adding two vectors means combining both their sizes and directions, not just their numbers.

What is the difference between a vector and a scalar?

A scalar is fully described by a single number (with a unit): 5 kg, 20°C, 60 mph. A vector needs a number and a direction to be fully described: 60 mph due north is a vector (a velocity), while 60 mph on its own is a scalar (a speed).

• Displacement (a vector) is the straight-line distance and direction from start to end. Distance (a scalar) is the total length travelled, whichever path was taken.
• Velocity (a vector) is speed in a given direction. Speed (a scalar) ignores direction.
• Force, acceleration, momentum and weight are vectors. Mass, energy, time and temperature are scalars.

A runner who does one full lap of a 400 m track has travelled a distance of 400 m, but their displacement is zero — they end up back where they started.

How do you add vectors?

To add vectors, draw them tip to tail and find the single arrow (the resultant) that reaches from the start to the final tip, as shown below.

Vectors add tip to tail: draw the first vector, then start the second vector from the tip (arrowhead) of the first. The resultant vector is the single arrow from the start of the first vector to the tip of the last one. This works for any number of vectors, in any order.

Worked example: two forces at right angles

A box is pulled by a 3 N force to the east and a 4 N force to the north, at the same time. What is the resultant force?

Because the two forces are at right angles, the resultant's magnitude comes from Pythagoras' theorem:

magnitude = √(3² + 4²) = √25 = 5 N

The direction is found with trigonometry: angle from east = tan⁻¹(4/3) ≈ 53.1°. So the resultant is 5 N at 53.1° north of east.

Adding vectors that aren't at right angles

When two vectors don't meet at 90°, split each one into horizontal and vertical components first (see below), add the components separately, then recombine. This always works, while the simple Pythagoras shortcut above only works for a right angle.

What are vector components?

Any vector can be split into a horizontal component and a vertical component that, added together, give back the original vector. For a vector of magnitude V at angle θ above the horizontal:

• horizontal component: Vₓ = V cos θ
• vertical component: Vᵥ = V sin θ

Worked example: resolving a force

A rope pulls a sledge with a force of 50 N at 30° above the horizontal. What are the horizontal and vertical components?

Vₓ = 50 × cos 30° = 50 × 0.866 ≈ 43.3 N

Vᵥ = 50 × sin 30° = 50 × 0.5 = 25 N

The horizontal component is what actually drags the sledge forward; the vertical component partly lifts it off the ground.

How do you find the magnitude of a vector?

The magnitude of a vector is its length, always a positive scalar, often written |v| or just v. For a vector given by its components (x, y), the magnitude of a vector comes straight from Pythagoras' theorem:

|v| = √(x² + y²)

For a 3D vector (x, y, z), extend the same idea: |v| = √(x² + y² + z²).

Worked example

A displacement vector has components (6 m, 8 m). Its magnitude is √(6² + 8²) = √100 = 10 m. Its direction is tan⁻¹(8/6) ≈ 53.1° from the x-axis.

What is the dot product of two vectors?

The dot product of two vectors turns them into a single number, as worked through below.

The dot product (or scalar product) combines two vectors into a single scalar number:

a · b = |a| |b| cos θ

where θ is the angle between the two vectors. It is also equal to the sum of the products of matching components: for a = (a₁, a₂) and b = (b₁, b₂), a · b = a₁b₁ + a₂b₂.

The dot product shows up throughout physics: work done = force · displacement, which is why a force applied at right angles to the direction of motion (θ = 90°, cos θ = 0) does zero work — pushing straight down on a trolley you're pulling forward does nothing to speed it up.

Worked example

Find the dot product of a = (3, 4) and b = (2, −1).

a · b = (3 × 2) + (4 × −1) = 6 − 4 = 2

Because the result is positive, the angle between the vectors is less than 90°. A dot product of zero means the vectors are perpendicular.

Common mistakes with vectors

• Adding magnitudes directly without regard to direction. 3 N and 4 N only add to 7 N if they point the same way; at right angles the resultant is 5 N, not 7 N.
• Forgetting displacement can be zero even after a long journey. Distance and displacement are different quantities measuring different things.
• Mixing up sin and cos when resolving components. The component along the angle uses cos; the component perpendicular to it uses sin — drawing the right-angled triangle first avoids guessing.
• Treating the dot product as a vector. It always produces a single number (a scalar), never another vector.

For the forces that vectors most often describe, see Newton's laws of motion. For the trigonometry behind resolving components, see the unit circle explained.

Frequently asked questions

What is a vector in maths?

A vector is a quantity that has both a magnitude (size) and a direction, usually represented as an arrow or as a set of components like (x, y). This is different from a scalar, which has only a size. Common vector quantities include displacement, velocity, acceleration and force.

How do you subtract vectors?

To subtract vector b from vector a, add a to the negative of b: a − b = a + (−b). The negative of a vector has the same magnitude but points in the opposite direction, so geometrically, reverse the arrow for b and then add it tip to tail onto a as usual.

What is a unit vector?

A unit vector is a vector with a magnitude of exactly 1, used to describe a pure direction without worrying about size. Any vector can be turned into a unit vector by dividing each of its components by its own magnitude; this process is called normalising the vector.

Can a vector have a negative magnitude?

No. Magnitude is always zero or positive, because it measures length, which can't be negative. A negative sign in front of a vector (like −v) reverses its direction; it does not make its magnitude negative — the magnitude of −v is the same as the magnitude of v.

Try it yourself

Open Vectors and build a vector by dragging its tip. Add a second vector tip to tail and watch the resultant vector appear automatically, then switch to component view to see the same vector broken into its horizontal and vertical parts.