Vectors

Some quantities need a direction as well as a size, and they do not add like numbers.

A plane points due north and flies at 300 mph.

There is a 50 mph wind from the west. The pilot has not turned; the nose is still pointing north.

pointed due north, and not going there 300 nose 50 wind actual path a heading and a speed, added — but not like numbers 300 plus 50 is not 350 when they point different ways

The plane still ends up east of where it aimed, because the air it is flying through is moving too.

Its real speed is not 350, and not 300 either. It is about 304, in a direction nobody set.

Two quantities that each carry a direction add tip-to-tail, not digit-to-digit. That is the whole idea of a vector, and it is why the answer is a triangle rather than a sum.

Adding vectors head to tail, and reading their components head to tail u v u + v components 4 across 3 up length 5 two ways to hold the same arrow

Components: the useful form

Any vector splits into a horizontal part and a vertical part, written ⟨ a, b ⟩.

From a magnitude and direction, the components are ⟨ rcosθ, rsinθ ⟩ — the same conversion as polar coordinates.

Going back, |v| = √a2+b2 and the direction comes from tanθ = ba.

Components are for arithmetic

Adding is trivial in components and awkward in magnitude-and-direction form. Convert first, add second, convert back last.

Adding tip to tail

Draw the second vector starting where the first one ends. The resultant runs from the very start to the very end.

In components it is simply ⟨ a1+a2, b1+b2 ⟩ — add the parts that point the same way.

Order does not matter: going east then north lands you where going north then east does.

Magnitudes do not add

|u| + |v| is almost never |u+v|. It is only equal when the two point the same way — otherwise the triangle is shorter than going round two sides.

Scalar multiplication

Multiplying by a number stretches the vector without turning it: 3v is three times as long, same direction.

A negative scalar reverses it, so -v has the same length pointing the opposite way.

Subtracting is adding the reverse: u-v means u+(-v).

Unit vectors

Dividing a nonzero vector by its own magnitude gives a vector of length 1 pointing the same way. The zero vector has no direction and cannot be normalised. That is how you keep a direction and throw away the speed.

The dot product

u · v = a1a2 + b1b2 — multiply matching components and add. The result is a number, not a vector.

It also equals |u||v|cosθ, which is what makes it useful: it measures how much two vectors agree.

For two nonzero vectors, a dot product of zero means cosθ = 0, so the vectors are perpendicular.

A one-line perpendicularity test

⟨ 3,4⟩ · ⟨ -4,3⟩ = -12+12 = 0, so those two are at right angles. No angles need to be calculated.

Navigation and force problems

Write every quantity in components, add them, then convert the resultant back to a speed and a bearing.

Bearings are measured clockwise from north, not anticlockwise from east — convert carefully or the answer is reflected.

When forces balance, the resultant is the zero vector, which is two equations: horizontal parts cancel, vertical parts cancel.

Draw the triangle first

A rough sketch tells you roughly where the answer should land. A component slip usually shows up as an answer pointing the wrong way entirely.

Worked examples

Adding in components

Find u+v and its magnitude, for u=⟨ 3,-2⟩ and v=⟨ -1,5⟩.

  1. Add matching components: ⟨ 3+(-1), -2+5 ⟩. Horizontal with horizontal, vertical with vertical.
  2. That gives ⟨ 2, 3 ⟩. The resultant, in component form.
  3. |⟨ 2,3⟩| = √4+9 = √13 ≈ 3.6. Pythagoras on the components.

⟨ 2,3 ⟩, magnitude √13

The plane and the wind

A plane flies north at 300 mph in a 50 mph wind from the west. Find the ground speed to one decimal place.

  1. North is ⟨ 0, 300 ⟩; a wind from the west pushes east, so ⟨ 50, 0 ⟩. Components first.
  2. The resultant is ⟨ 50, 300 ⟩. Add the parts.
  3. √2500+90000 = √92500 ≈ 304.1 mph. Not 350 — the wind is sideways, so most of it is wasted.

about 304.1 mph

Testing for perpendicular

Are ⟨ 6, -2 ⟩ and ⟨ 1, 3 ⟩ perpendicular?

  1. Compute the dot product: 6(1) + (-2)(3). Multiply matching components.
  2. = 6 - 6 = 0. The parts cancel exactly.
  3. A zero dot product means cosθ = 0, so yes — they meet at 90°. No angle had to be found.

Yes

Words to know

Vector
A quantity with both magnitude and direction.
Scalar
A quantity with size only, such as mass or temperature.
Magnitude
The length of the vector, written |v|.
Component form
A vector written as its horizontal and vertical parts, ⟨ a, b ⟩.
Resultant
The single vector that two or more vectors add up to.

Mistakes to watch for

Adding magnitudes instead of vectors. A 300 mph plane in a 50 mph crosswind does not travel 350 mph; it travels about 304. Only vectors pointing the same way add their lengths.

Treating the dot product as a vector. u · v is a single number. An answer written as an ordered pair has confused it with something else.

Getting the wind direction backwards. A wind from the west blows towards the east, so its component is positive. Reversing it puts the plane on the wrong side of its course.

Mixing bearings with standard angles. Bearings run clockwise from north; the usual θ runs anticlockwise from east. Using one convention’s number in the other’s formula reflects the answer.

Forgetting that a negative scalar reverses direction. -2v is twice as long and points the opposite way, not merely twice as long.

Hamilton spent years looking for a way to multiply points in three dimensions, and carved the answer into a Dublin bridge in 1843 when it finally came to him.

Force, velocity, acceleration and momentum are all vectors. Mass, temperature and time are not — they have size and nothing else.

Every 3D game moves its objects by adding vectors thousands of times a second. The arithmetic in this unit is literally what the graphics card does.

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