Vector representation — magnitude, direction, components: Question bank

Definition Notation used on this whole page

Before any trap, lock in the symbols so no line surprises you:

$A$ — the arrow itself (bold-with-arrow means "this is a vector, it carries a direction").
$A = ∣ A ∣$ — the magnitude, the plain length of the arrow. Always $A \geq 0$ (a length can't be negative).
$A_{x}, A_{y}$ — the components: how far the arrow reaches along the horizontal (x) and vertical (y) axes. These can be negative (pointing left/down).
$\hat{i}, \hat{j}$ — unit vectors of length 1 along $+ x, + y$ ; we write $A = A_{x} \hat{i} + A_{y} \hat{j}$ .
$θ$ — the direction angle, measured anticlockwise (CCW) from the +x axis. This one convention is the hinge of every cos/sin and quadrant rule below — the x-side is adjacent to $θ$ , so it gets cosine.
$\hat{A}$ — the "hat" means a unit vector: same direction as $A$ but length exactly 1, built by $\hat{A} = A / A$ .

Figure — Vector representation — magnitude, direction, components

True or false — justify

True or false: A vector's magnitude can be negative if it points down-left.

False — magnitude is

A = A_{x}^{2} + A_{y}^{2} \geq 0

always; the "down-left" news lives entirely in the signs of the components and the angle, never in the size.

True or false: Two vectors with the same magnitude are the same vector.

False — a vector needs both magnitude and direction; two arrows can be equally long yet point different ways, so they are different vectors.

True or false: The x-component always uses cosine, no matter how the angle is drawn.

False — cosine goes with whichever axis the angle is measured from. Here

θ

is CCW from +x, so the x-side is adjacent and gets cosine; if

θ

were measured from the +y axis, the y-side would be adjacent instead.

True or false:

∣ A + B ∣ = ∣ A ∣ + ∣ B ∣

for any two vectors.

False — arrows can partly cancel, so the resultant is usually shorter than the sum; equality holds only when

A

and

B

point the same way (parallel).

True or false: A unit vector has magnitude 1 but no direction.

False — a unit vector

\hat{A}

is pure direction; it keeps the direction of the original and throws away only the size, so its magnitude is exactly 1.

True or false: If a vector has

A_{x} = 0

, its angle is undefined.

False — it points straight along

\pm y

, so

θ = 90°

(if

A_{y} > 0

) or

270°

(if

A_{y} < 0

); it's the formula

tan^{- 1} (A_{y} / A_{x})

that breaks by dividing by zero, not the direction itself.

True or false: Multiplying a vector by

- 1

leaves its magnitude unchanged.

True — every component flips sign, but

(- A_{x})^{2} + (- A_{y})^{2} = A_{x}^{2} + A_{y}^{2}

; only the direction reverses (rotates by

180°

True or false: A vector at

30°

and one at

390°

are different vectors.

False —

390° = 30° + 360°

, one full turn back to the same direction, so (with equal magnitude) they are identical arrows.

Spot the error

Spot the error: "

A = - 3 \hat{i} + 4 \hat{j}

, so

θ = tan^{- 1} (4/ - 3) = - 53°

The raw

tan^{- 1}

is right but incomplete:

A_{x} < 0

means the arrow points into Quadrant II, so add

180°

to get the true angle

127°

Spot the error: "For

θ = 60°

from +x,

A_{x} = A sin 60°

Sine and cosine are swapped — the angle is measured from the x-axis, so the x-side is adjacent and uses cosine:

A_{x} = A cos 60°

Spot the error: "A force is 50 N at

30°

below +x, so

F_{y} = 50 sin 30° = + 25

N."

Below the axis means

θ = - 30°

, so

F_{y} = 50 sin (- 30°) = - 25

N; the sign must be negative because the component points downward.

Spot the error: "

\hat{A} = A / A

, and since dividing changes each component, the direction changes too."

Direction is untouched — every component shrinks by the same factor

A

, so their ratio (and hence the slant) stays fixed; only the length becomes 1.

Spot the error: "

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

, and for

- 3 \hat{i} + 4 \hat{j}

that's

- 3^{2} + 4^{2} = 7

The squares kill the sign:

(- 3)^{2} = + 9

, not

- 9

; so

A = 9 + 16 = 5

. Missing the parentheses on the negative component is the slip.

Spot the error: "Two vectors add to give

∣ A + B ∣ = 7

from

∣ A ∣ = 3, ∣ B ∣ = 4

This assumes they're parallel;

7

is the maximum resultant. The minimum is

∣3 - 4∣ = 1

, which happens when they're antiparallel and the shorter one cancels part of the longer — so the true answer sits anywhere in

[1, 7]

depending on the angle between them.

Spot the error: "The calculator gives

tan^{- 1}

, so that's THE angle of the vector."

tan

repeats every

180°

, so two opposite directions share one ratio; a plain

tan^{- 1}

only returns

(- 90°, 90°)

. The clean fix is the two-argument arctan,

atan2 (A_{y}, A_{x})

, which is fed both signs separately and so returns the correct angle in all four quadrants over

(- 180°, 180°]

Why questions

Why does the x-component use cosine and not sine (angle CCW from +x)?

Because

θ

is measured from the x-axis, making the x-side the one touching the angle (adjacent), and cosine = adjacent/hypotenuse.

Why must you check quadrants after

tan^{- 1} (A_{y} / A_{x})

Because

A_{y} / A_{x}

and

(- A_{y}) / (- A_{x})

give the identical ratio, so tan cannot distinguish an arrow from its exact opposite; only the signs of the components reveal which of the two it is.

Why does

atan2 (A_{y}, A_{x})

solve the quadrant ambiguity automatically?

Because it takes

A_{y}

and

A_{x}

as two separate inputs instead of pre-dividing them into one ratio, it can read each sign individually and place the angle in the correct quadrant — the sign information that a single ratio throws away is exactly what it keeps.

Why can't a single number (scalar) represent a velocity fully?

Velocity has both "how fast" and "which way"; a lone number holds only one piece, so you'd lose the direction — that's why we need an arrow with two pieces of news.

Why do components add like ordinary numbers but whole arrows don't?

Components lie along fixed perpendicular axes, so x's only interact with x's and y's with y's; slanted arrows mix directions, so they can partially cancel and need resolving first.

Why is dividing by

A

the right way to build a unit vector?

Dividing by the length rescales the arrow to length 1 while preserving the ratio of its parts, giving pure direction with the size normalized away.

Why does magnitude stay

\geq 0

even for a vector pointing into Quadrant III?

The formula squares each component, erasing all sign information before taking a positive square root — so magnitude is a length, and lengths are never negative.

Edge cases

Edge case: What is the direction angle of the zero vector

0

Undefined — it has no length and points nowhere, so no meaningful angle exists (and

tan^{- 1} (0/0)

is indeterminate).

Edge case:

A = 5 \hat{i} + 0 \hat{j}

— what is

θ

, and does the formula work?

θ = 0°

; the formula

tan^{- 1} (0/5) = 0°

works cleanly and no quadrant fix is needed since

A_{x} > 0

Edge case:

A = 0 \hat{i} - 4 \hat{j}

— what is

θ

It points straight down, so

θ = 270°

(or

- 90°

); the formula

tan^{- 1} (- 4/0)

divides by zero, so you must read the direction geometrically instead.

Edge case:

A = - 2 \hat{i} + 0 \hat{j}

— what does the naive

tan^{- 1}

give and what's true?

tan^{- 1} (0/ - 2) = 0°

, but the arrow points along

- x

, so the true angle is

180°

; the

A_{x} < 0

rule adds

180°

and fixes it.

Edge case: A vector at exactly

θ = 90°

— what are its components?

A_{x} = A cos 90° = 0

and

A_{y} = A sin 90° = A

; it lies entirely along +y, confirming the formulas handle the boundary correctly.

Edge case: Can a unit vector exist for the zero vector?

No — you'd divide by

A = 0

, which is undefined; the zero vector has no direction to normalize.

Edge case: Two vectors have equal magnitude and add to zero — what's their relationship?

They are exact opposites (antiparallel), each the negative of the other, so their components cancel component-by-component to give

0

Connections

Scalars vs Vectors — the "why one number isn't enough" traps.
Vector Addition — Triangle & Parallelogram Law — the "don't add magnitudes" trap.
Resolution of Forces — where the cos/sin swap bites hardest.
Dot Product and Cross Product — sign and direction subtleties continue.
Projectile Motion — components of velocity in action.
Unit Vectors i, j, k — pure-direction reasoning.