🏹Vectors

"Walk 3 kilometres forward" leaves something out — which way? 3 km east and 3 km north land far apart. A quantity carrying both size and direction, like displacement, is a vector; one with only size is a scalar, like 3 kg or 20 dollars. Speed reports size alone — a scalar; attach a direction and it is promoted to velocity, a vector.

Components: split the arrow into two steps

In the coordinate plane, a vector can be written as a pair $(3, 4)$: 3 east first, then 4 north. This pair is called the components — one slanted flight split into "a horizontal walk + a vertical walk".

x3y-2

(3, -2) → Quadrant IV

Walk x steps horizontally, then y steps vertically — the ordered pair (x, y) pins down exactly one point.

Drag the sliders and follow the dashed path, horizontal first then vertical: from the origin to $(3, -2)$ means "3 across, −2 up (that is, 2 down)". A vector starting at the origin is a position vector — the coordinates themselves are its components.

Adding: nose to tail

3 east and 4 north, then 1 more east and 2 more north — where are you now? Vector addition just adds the components separately:

$(3, 4) + (1, 2) = (4, 6)$

Geometrically it is nose to tail: the tail of the second arrow meets the head of the first, and the arrow from the very start to the very finish is the sum. Order does not matter — east then north lands exactly where north then east does.

Scaling and magnitude

To stretch a vector to twice its length, multiply both components by 2:

$2 \times (-1, 5) = (-2, 10)$

A negative factor turns it around: $-1 \times (3, 4) = (-3, -4)$ — same length, opposite direction.

How long is the arrow itself? Treat the components as the two legs of a right triangle and call on the Pythagorean theorem:

$|(3, 4)| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$

3 km east then 4 km north is exactly 5 km as the crow flies — the old 3-4-5 trio again. One more: $|(-6, 8)| = \sqrt{36 + 64} = \sqrt{100} = 10$.

The translation viewpoint

A vector's most distinctive temper: it cares how far you travelled, not where you started. Drawn at the origin or anywhere else, $(3, 4)$ is the same arrow. That is why the translation of a figure takes one sentence: add the same vector to every point — $(2, 3)$ slides the whole figure 2 right and 3 up, shape and size untouched. Behind every sliding figure in Transformations stands this idea.

Check yourself

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Quick quiz

↺ Try again

1. 1. What is the magnitude of the vector (3, 4)?

   5712

2. 2. What is (2, -1) + (1, 3)?

   (3, 2)(3, 4)(2, 2)

3. 3. Shrink the vector (2, 5) to half its length. What do you get?

   (1, 2.5)(4, 10)(2, 2.5)