Vector representation — magnitude, direction, components: Foundations

This page assumes you know nothing. Before you can read the parent note, you need to recognise a small handful of symbols and pictures. We build them one at a time, and each one uses only the ones above it. If you have never seen an arrow used as a number, start here at line one.

1. The arrow itself — what a vector looks like

Forget formulas. A vector is a drawn arrow: it has a starting tail, a pointed head, a length, and a slant.

Figure — Vector representation — magnitude, direction, components

Plain-words meaning: "this much, in this direction."
The picture: the orange arrow above. Its length is the "this much"; the way it tips is the "this direction."
Why the topic needs it: forces, velocities and displacements all push a specific amount a specific way. The arrow is the only object that stores both at once.

2. The playground — axes, origin, +x and +y

To turn a picture into numbers we need a ruler grid to draw the arrow on.

Plain-words meaning: graph paper with a marked centre and two marked "positive" directions — right ( $+ x$ ) and up ( $+ y$ ).
The picture: the navy cross above; the magenta dot is the origin; the orange arrow labelled $+ x$ points right, the violet arrow labelled $+ y$ points up.
Why the topic needs it: we always draw a vector starting from the origin. Then "how far right" (along $+ x$ ) and "how far up" (along $+ y$ ) become simple readings off the grid. Without a fixed grid, "which way" has no meaning.

3. The angle $θ$ — the "which way" as a number

Plain-words meaning: how far you have rotated away from "pointing right."
The picture: the violet wedge above, opening from the $+ x$ arrow up to the vector.
Why the topic needs it: direction is half the vector's news. $θ$ is that half, written as a number so we can compute with it.

So a vector's two-number description is $(A, θ)$ — length and angle. This is called the polar form.

4. The right triangle — the machine hiding inside the arrow

Now the key move. Drop a straight line straight down from the arrow's head to the x-axis. Look what appears:

The picture: the orange arrow is the hypotenuse; the horizontal navy leg runs along the x-axis; the vertical magenta leg is the dropped line.
Why the topic needs it: this triangle is what lets us swap between the two descriptions. The hypotenuse is the arrow's length $A$ ; the corner angle is $θ$ ; the two legs are the numbers we're after in the next section.

Notice the horizontal leg answers "how far right did the arrow go?" and the vertical leg answers "how far up?" Those two answers are the components.

5. The trig ratios — turning a slant into a ratio

Before we can write any formula for the components we must build the tools that connect the angle $θ$ to the sides of the triangle. Those tools are the trigonometric ratios. This section comes before the component formulas on purpose — you cannot use $cos$ and $sin$ until they mean something.

Why $tan^{- 1}$ matters: going from the two legs back to an angle, we only have their ratio. $tan^{- 1}$ is the tool that reads an angle out of a ratio.

6. Components — the "how far right / how far up" numbers

The picture: in the triangle figure (Section 4), $A_{x}$ is the length of the navy horizontal leg, $A_{y}$ the length of the magenta vertical leg.
Why the topic needs it: slanted arrows are hard to add. Horizontal and vertical pieces add like ordinary numbers on a ruler. Components are the arrow rewritten as two easy movements ("go right, then go up").
Signs: if the arrow reaches left, $A_{x}$ is negative; if it reaches down, $A_{y}$ is negative. The sign is a direction flag.

Now that $cos$ and $sin$ exist (Section 5), we can write the bridge. Take $cos θ = A_{x} / A$ and $sin θ = A_{y} / A$ off the triangle, then multiply both sides by the hypotenuse $A$ :

7. Every quadrant and every degenerate case

The single pair $A_{x} = A cos θ, A_{y} = A sin θ$ already handles every direction, including the flat and straight-up cases. Look at how the signs flip as the arrow rotates.

8. Pythagoras — recovering the length

Plain-words meaning: the straight-line length equals the diagonal of the "go right, go up" rectangle.
The picture: the same right triangle — the two legs and the hypotenuse.
Why the topic needs it: it recovers the magnitude $A$ from the two components. It is the "go East then North → straight-line distance" rule. Note it also works in the degenerate cases: if $A_{y} = 0$ then $A = A_{x}^{2} = ∣ A_{x} ∣$ , as expected.

9. Unit vectors $\hat{i}, \hat{j}$ — the tiny signposts

First we must say exactly what the little hat means, because a newcomer cannot guess it.

Plain-words meaning: a "one step East" arrow and a "one step North" arrow.
Why the hat means length one: because that is precisely what the hat operator does — divide an arrow by its own length and you are left with a length-one copy. (You can check with Pythagoras: an arrow of components $(0.6, 0.8)$ has length $0. 6^{2} + 0. 8^{2} = 1$ , so it earns a hat.)
Why the topic needs it: to write the component form $A = A_{x} \hat{i} + A_{y} \hat{j}$ , which reads literally as "go $A_{x}$ steps East, then $A_{y}$ steps North." The $\hat{i}, \hat{j}$ tell each number which direction it applies to.

10. The full symbol dictionary

Recall Every symbol on one card (click to reveal)

$A$ ::: a vector — an arrow with size and direction. $A$ or $∣ A ∣$ ::: magnitude — the arrow's length, always $\geq 0$ . $θ$ ::: direction angle, anticlockwise from $+ x$ . $+ x, + y$ ::: the positive grid directions — right and up. $A_{x}$ ::: x-component — how far right (adjacent leg); negative if the arrow points left. $A_{y}$ ::: y-component — how far up (opposite leg); negative if the arrow points down. $cos θ$ ::: adjacent ÷ hypotenuse. $sin θ$ ::: opposite ÷ hypotenuse. $A_{x} = A cos θ$ ::: x-component from polar form (x-side is adjacent → cosine). $A_{y} = A sin θ$ ::: y-component from polar form (y-side is opposite → sine). $tan θ$ ::: opposite ÷ adjacent = steepness. $tan^{- 1}$ ::: the undo of tan — "which angle gives this ratio?"; undefined when $A_{x} = 0$ . ::: square root — undoes squaring; used in $A = A_{x}^{2} + A_{y}^{2}$ . $^$ ::: the hat operator — "shrink to length 1"; marks a unit vector. $\hat{i}, \hat{j}$ ::: length-1 signpost arrows along $+ x$ and $+ y$ .

Prerequisite map

Equipment checklist

Test yourself — you are ready for the parent note only if you can answer each without peeking.

Can I point to the magnitude and the direction on a drawn arrow?

Length = magnitude; slant = direction.

Do I know where

θ

is measured from and which way it turns?

From

+ x

, anticlockwise (positive).

Do I know what

+ x

and

+ y

mean?

+ x

= right along the x-axis;

+ y

= up along the y-axis.

Can I say which quadrant an arrow is in from

θ

0

–

90°

, II:

90

–

180°

, III:

180

–

270°

, IV:

270

–

360°

Can I drop a perpendicular from an arrow's head to make a right triangle?

Yes — vertical line to the x-axis; hypotenuse = arrow, legs = components.

Which leg is adjacent to

θ

, which is opposite?

Adjacent = the leg touching

θ

(horizontal); opposite = the vertical leg.

Do I know

cos θ

and

sin θ

as side ratios?

cos = adj / hyp

sin = opp / hyp

Can I write the two component formulas?

A_{x} = A cos θ

and

A_{y} = A sin θ

Do I know why the formulas give the right signs in every quadrant?

Because

cos

and

sin

go negative on their own past

90°

and

180°

What happens at

θ = 90°

to the components?

A_{x} = 0

A_{y} = A

— the triangle flattens to a vertical line.

What does

tan^{- 1}

do, and when is it undefined?

Returns the angle with a given steepness; undefined when

A_{x} = 0

Can I state Pythagoras and the magnitude formula?

A^{2} = A_{x}^{2} + A_{y}^{2}

, so

A = A_{x}^{2} + A_{y}^{2}

What does the hat

^

mean?

"Length exactly 1" — it marks a unit vector.

What do

\hat{i}

and

\hat{j}

mean?

Length-1 arrows along

+ x

and

+ y

— pure direction signposts.

Is magnitude ever negative?

Never;

A \geq 0

. Sign info lives in components.

Connections

Scalars vs Vectors — why one number can't hold size and direction.
Unit Vectors i, j, k — the signposts $\hat{i}, \hat{j}$ built here, extended to 3D.
Vector representation — magnitude, direction, components — the parent topic these foundations feed.
Vector Addition — Triangle & Parallelogram Law — where components pay off.
Resolution of Forces — trig ratios applied to forces.
Dot Product and Cross Product — built on components.
Projectile Motion — splitting velocity into $v_{x}, v_{y}$ .