Vectors: Magnitude, Direction and Components

Quantities that point

Some quantities are a single number. A temperature of $70°$F, a mass of $3$ kilograms, a speed of $60$ miles per hour: each is a scalar, a size and nothing more.

Others need a direction to mean anything. A plane flying $60$ miles per hour north and one flying $60$ miles per hour east end up in different states. A push of $10$ pounds to the left undoes a push of $10$ pounds to the right. These are vectors: each has a magnitude, its size, and a direction.

Scalar	Vector
speed: $60$ mph	velocity: $60$ mph north
distance: $5$ miles	displacement: $5$ miles southwest
mass	force

Drawing and naming a vector

A vector is drawn as an arrow, a directed segment. Its length is the magnitude and it points in the vector’s direction. It starts at an initial point and ends at a terminal point.

Several notations are in use, and they all mean the same thing:

Notation	Meaning
$\mathbf{v}$ or $\vec{v}$	the vector named $v$
$\overrightarrow{AB}$	the vector from $A$ to $B$
$\lVert\mathbf{v}\rVert$ or $\lvert\mathbf{v}\rvert$	its magnitude
$\langle a, b\rangle$	the vector moving $a$ across and $b$ up

Components

The arrow from $A(1, 2)$ to $B(5, 5)$ moves $4$ right and $3$ up. Those two numbers are its components, found by subtracting the start from the end:

$$\overrightarrow{AB} = \langle 5 - 1,\; 5 - 2\rangle = \langle 4, 3\rangle$$

The magnitude is the length of the arrow, from the distance formula:

$$\lVert\langle 4, 3\rangle\rVert = \sqrt{4^2 + 3^2} = 5$$

Why the components do not depend on where the arrow starts

A vector records a change: move this far across and this far up. It says nothing about where the move begins. So every arrow with the same length and direction is the same vector, wherever it is drawn.

Subtraction is what makes this true. Sliding an arrow adds the same amount to both its start and its end, and those amounts cancel when one is subtracted from the other. The components survive every slide, which is why they, and not the endpoints, describe the vector.

Magnitude and direction

A vector can also be described by its magnitude $r$ and its direction angle $\theta$, measured counterclockwise from the positive $x$-axis. The two descriptions convert like polar and rectangular coordinates:

$$\langle a, b\rangle = \langle r\cos\theta,\; r\sin\theta\rangle \qquad r = \sqrt{a^2 + b^2}$$

For the direction, $\tan\theta = \tfrac{b}{a}$ — but the calculator’s answer needs checking against the quadrant the arrow points into.

Velocity

A plane flies at $400$ miles per hour, $30°$ north of east. Its velocity vector splits into an eastward part and a northward part:

$$\mathbf{v} = \langle 400\cos 30°,\; 400\sin 30°\rangle \approx \langle 346.4,\; 200\rangle$$

Each hour the plane moves about $346.4$ miles east and $200$ miles north. The components answer questions the speed alone cannot, such as how soon the plane crosses a line of latitude $600$ miles to the north: $600 \div 200 = 3$ hours.

Worked examples

Common mistakes

Practice problems