Speak Maths · Last school years

Vectors: arrows with size and direction

A vector is not just a number: it has a size and a direction. A few signs tell us how to write one, measure it and multiply two of them.

Draft: written by an AI helper and not yet checked by a teacher.

The alphabet

Read each sign aloud, the way it is written under it.

$v$ Say: vector vMeans: an arrow with a length and a direction; by hand we draw a small arrow over the letter
$v = (v_{1}, v_{2})$ Say: the vector v with components v one and v twoMeans: the same arrow in bold type; its components say how far it goes along each axis
$∥ v ∥$ Say: the norm of v, or the length of vMeans: how long the arrow is: a single number
$u \cdot v$ Say: u dot vMeans: a number, not an arrow; it is zero when the two arrows meet at a right angle
$u \times v$ Say: u cross vMeans: a new arrow at right angles to both; its length is the area of the parallelogram they span
$u^= \frac{u}{∥ u ∥}$ Say: u hat equals u over the norm of uMeans: the unit vector: same direction as u, but length one

The story: The boat and the current

Subtitles:

Tap a line to see its words.

$b = (0, 4)$ A boat in still water goes 4 metres per second straight across the river, north. East is x, north is y.
$c = (3, 0)$ But the river itself flows east at 3 metres per second.
$v = b + c = (3, 4)$ The boat feels both at once: its real velocity is the sum of the two arrows.
$∥ v ∥ = 5$ The real speed is the length of that arrow: 5 metres per second, the long side of a 3, 4, 5 triangle.
$v^= (0.6, 0.8)$ To keep only the direction, divide by the length: 0.6 east and 0.8 north, a unit arrow.
$b \cdot c = 0$ The boat’s arrow and the current’s arrow meet at a right angle, so their dot product is zero: the current does not push the boat any further north.
$b \times c = (0, 0, - 12)$ With a third direction pointing up, their cross product points straight down, with length 12: the area of the parallelogram they span.
$10 v = (30, 40)$ The river is 40 metres wide, so the crossing takes 10 seconds. The boat travels ten times its velocity: 30 metres east and 40 north.
$∥ 10 v ∥ = 50$ It covers 50 metres, though the river is only 40 wide, and lands 30 metres downstream of where it aimed.

In words this story is 947 letters long; in symbols, 85.

Read it

What does it say?

$∥ v ∥ = 5$

$u \cdot v = 0$

$∥ u^∥ = 1$

$w = u \times v$

$u \cdot v = ∥ u ∥ ∥ v ∥$

$w = (2, - 5)$

$∥ u \times v ∥ = 0$

$a + b = 0$

$∥ 3 v ∥ = 3 ∥ v ∥$

$v^= (0, 1)$

$u \cdot v < 0$

$F \cdot d = 12$

Write it

Which notation says this?

The length of the vector v.

The vectors u and v meet at a right angle.

The unit vector in the direction of u is u divided by its length.

The arrow v, going 3 along x and 4 along y.

The area of the parallelogram spanned by u and v.

The sum of two arrows a and b, each with a small arrow drawn over its letter.

The length of the arrow (5, 12) is 13.

The dot product of (1, 2) and (3, 4) is 11.

The unit arrow in the direction of (3, 4): divide the arrow by its length, 5.

The cross product of u and v is a new arrow w, at right angles to both.

Two arrows point exactly the same way: their dot product is the product of their lengths.

The arrow v, scaled down to length one.

Build it

Tap the tiles in order. One tile does not belong.

u hat equals u over the norm of u.

u dot v equals zero.

The length of the arrow (3, 4) is 5.

u cross v equals w.

Vector a plus vector b equals vector c.

The dot product of (2, 1) and (1, 3) equals 5.

Challenges

Creative puzzles that use every sign of this chapter. Take your time; there is no clock.

1 Four arrows in the plane

Here u = (3, 4), v = (4, −3), w = (−3, −4), z = (1, 0). Judge each sentence: true or false?

	$u$	$v$	$w$
$u$		✓
$v$	✓		✓
$w$		✓
$z$

A tick means the row’s arrow and the column’s arrow meet at a right angle, so their dot product is zero.

$u \cdot v = 0$
$u \cdot w = 0$
$∥ v ∥ = 7$
$u^= (0.6, 0.8)$
$u + v = (7, 1)$
$∥ u ∥ = ∥ w ∥$

2 Six signs, six meanings

These lines look alike but say very different things. Pair each with its meaning.

Tap a line, then its meaning.

3 Length of an arrow

This should find the length of the arrow (3, 4). One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

4 Make them meet at a right angle

Which value of k makes the two arrows perpendicular?

$u = (2, 3), v = (3, k), u \cdot v = 0$

5 The parallelogram

Two arrows along the x and y directions span a rectangle. How long is their cross product?

$u = (3, 0, 0), v = (0, 2, 0), ∥ u \times v ∥ = ?$

6 The rule of right angles

Write in symbols: two arrows are perpendicular exactly when their dot product is zero. Then compare.

Your version

7 Shrink to one

Dividing an arrow by its own length gives a unit arrow. Write in symbols that this arrow has length one. Then compare.

Your version

8 Four winds

Four arrows: a = (2, 0), b = (0, 3), c = (−2, 0), d = (1, 1). Judge each sentence: true or false?

	$a$	$b$	$c$	$d$
$a$	✓		✓
$b$		✓
$c$	✓		✓
$d$				✓

A tick means the row’s arrow and the column’s arrow are parallel (one is a multiple of the other), so their cross product is the zero vector.

$a \times c = 0$
$a + c = 0$
$a \cdot b = 0$
$∥ d ∥ = 2$
$b \times d = 0$
$∥ b \times d ∥ = 3$

9 Number or arrow?

Each line gives either a number or a new arrow. Pair each with what it gives.

Tap a line, then its meaning.

10 Dot, not pair

This should compute the dot product of (2, 3) and (4, 1). One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

11 Make it length one

This should find the unit arrow in the direction of v. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

12 The ladder

A ladder reaches 8 along the ground and 15 up a wall. How long is it?

$v = (8, 15), ∥ v ∥ = ?$

13 Rowing into the current

A boat rows with velocity b while the current pushes with c. How fast does the boat really move?

$b = (3, 4), c = (- 3, 0), ∥ b + c ∥ = ?$

14 Two axes make the third

The unit arrows along x and y are crossed. Which arrow do you get?

$u = (1, 0, 0), v = (0, 1, 0), u \times v = ?$

15 Pulling the kite

A string pulls with force F while the kite moves along d. Write in symbols that the work done is the dot product of force and displacement. Then compare.

Your version

16 Two legs of a journey

A journey has two legs, a then b. Write in symbols: the whole displacement is their sum, and its length is never more than the two leg lengths together. Then compare.

Your version

Tell your own

Tell a two-line story of something pushed two ways at once, a ball or a swimmer or a leaf, in signs only: two arrows, their sum, its length.

Kept on this device only.

Vectors: arrows with size and direction

The alphabet

The story: The boat and the current

Read it

Write it

Build it

Challenges

1 Four arrows in the plane

2 Six signs, six meanings

3 Length of an arrow

4 Make them meet at a right angle

5 The parallelogram

6 The rule of right angles

7 Shrink to one

8 Four winds

9 Number or arrow?

10 Dot, not pair

11 Make it length one

12 The ladder

13 Rowing into the current

14 Two axes make the third

15 Pulling the kite

16 Two legs of a journey

Tell your own

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