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Understanding
Vector Quantities

Learn how vectors capture both magnitude and direction: represent them as directed line segments, and master the notation \(\vec{v}\), \(|\vec{v}|\), and \(\|\vec{v}\|\).

Precalculus

What Is a Vector?

A vector is a mathematical object that has two essential properties: magnitude (how much) and direction (which way). Unlike ordinary numbers that sit on a number line, vectors live in space and point somewhere.

Magnitude
The length or size of the vector: how far, how fast, how strong.
Direction
The orientation in space: north, 30° above horizontal, toward a target.

Scalars vs. Vectors

A scalar is a quantity described by a single number (magnitude only). A vector requires both magnitude and direction. This is the most fundamental distinction in physics and mathematics.

Scalar Examples
Temperature (\(72^\circ\text{F}\)), mass (\(5\text{ kg}\)), speed (\(60\text{ mph}\)), time (\(3\text{ s}\)), energy (\(100\text{ J}\))
Vector Examples
Velocity (\(60\text{ mph east}\)), force (\(10\text{ N upward}\)), displacement (\(5\text{ m at }30^\circ\))

Vectors as Directed Line Segments

We represent a vector geometrically as an arrow. The arrow's length encodes the magnitude and the arrow's tip shows the direction.

Initial Point Terminal Point magnitude = length direction
\(\theta\)

Vector Notation and Symbols

Vectors have several standard notations. Recognizing each one is essential for reading textbooks, exams, and scientific papers.

\(\vec{v}\)
Arrow notation: a small arrow over the letter. Most common in precalculus and physics.
\(\mathbf{v}\)
Boldface notation: used in printed textbooks and linear algebra. Handwritten as \(\vec{v}\).
\(\overrightarrow{AB}\)
Endpoint notation: vector from point \(A\) (initial) to point \(B\) (terminal).

Magnitude: \(|\vec{v}|\) and \(\|\vec{v}\|\)

The magnitude of a vector is its length: a non-negative scalar. Two equivalent notations are used interchangeably:

\(|\vec{v}|\)
Single bars: common shorthand, especially in physics and precalculus texts.
\(\|\vec{v}\|\)
Double bars (norm): more formal, standard in linear algebra and advanced courses.

Formal Definition of a Vector

x y
\(A(1, 1)\)
\(B(5, 4)\)
horizontal: \(4\)
vertical: \(3\)
\(\|\vec{v}\| = 5\)
 1 2 3 4 5 1 2 3 4

When Are Two Vectors Equal?

Two vectors are equal if and only if they have the same magnitude and the same direction. Position does not matter: you can slide a vector anywhere without changing it.

\(\vec{v}\)
\(\vec{w}\)
 = Same magnitude, same direction Different position: still equal

Speed vs. Velocity

The most classic example of the scalar/vector distinction. These two words are not interchangeable.

Speed (Scalar) Velocity (Vector)
What it tells you How fast How fast and which way
Example \(60\text{ mph}\) \(60\text{ mph north}\)
Can be negative? No (always \(\ge 0\)) Components can be negative
Notation \(v\) (italic, no arrow) \(\vec{v}\) (arrow or bold)

How to Represent a Vector

1
Identify the initial and terminal points. Label them \(A\) (tail) and \(B\) (tip).
2
Find the component form. Subtract coordinates: \(\overrightarrow{AB} = \langle x_B - x_A,\; y_B - y_A \rangle\).
3
Calculate the magnitude. Apply the formula \(\|\vec{v}\| = \sqrt{a^2 + b^2}\).
4
Determine the direction. Use \(\theta = \tan^{-1}\!\left(\dfrac{b}{a}\right)\), adjusting for the correct quadrant.
5
Draw the directed line segment. Arrow from initial to terminal point, length proportional to magnitude.

Example 1: Finding Components

A vector goes from \(P(2, 1)\) to \(Q(6, 4)\). Find the component form and magnitude.

x y 1 2 3 4 5 6 1 2 3 4
\(P(2,1)\)
\(Q(6,4)\)
\(4\)
\(3\)

Example 2: Direction Angle

For the vector \(\vec{v} = \langle 4, 3 \rangle\), find the direction angle \(\theta\) measured from the positive \(x\)-axis.

x y O
\((4, 3)\)
\(\theta\)

Example 3: Quadrant Adjustment

Find the magnitude and direction of the vector from \(A(3, 5)\) to \(B(-1, 8)\).

1
Components: \(\overrightarrow{AB} = \langle -1-3,\; 8-5\rangle = \langle -4, 3 \rangle\)
2
Magnitude: \(\|\overrightarrow{AB}\| = \sqrt{(-4)^2 + 3^2} = \sqrt{16+9} = \sqrt{25} = 5\)
3
Reference angle: \(\tan^{-1}\!\left(\frac{|3|}{|-4|}\right) = \tan^{-1}\!\left(\frac{3}{4}\right) \approx 36.87^\circ\)
4
Quadrant II (\(x < 0, y > 0\)): \(\theta = 180^\circ - 36.87^\circ \approx\) \(143.13^\circ\)

Special Vectors

Zero Vector \(\vec{0}\)
Components \(\langle 0,0 \rangle\). Magnitude is 0. It has no defined direction: the only vector without one.
Unit Vector \(\hat{v}\)
A vector with magnitude exactly 1. It encodes direction only. Found via: \(\hat{v} = \frac{\vec{v}}{\|\vec{v}\|}\)

Common Mistakes to Avoid

✕ Confusing Scalars and Vectors
Writing "\(\text{velocity} = 60\text{ mph}\)" without direction. Velocity is a vector: always include direction.
✕ Subtracting Points Backwards
Computing \(\langle x_A - x_B, y_A - y_B \rangle\) instead of terminal minus initial. This reverses the direction.
✕ Wrong Quadrant for \(\theta\)
Using \(\tan^{-1}\!\left(\frac{b}{a}\right)\) without checking component signs. The calculator only returns values in \((-90^\circ, 90^\circ)\).

Vector Reference Summary

Concept Notation Formula / Meaning
Vector \(\vec{v}\) or \(\mathbf{v}\) Magnitude + direction
Component form \(\langle a, b \rangle\) \(\langle x_B - x_A,\; y_B - y_A \rangle\)
Magnitude \(\|\vec{v}\|\) or \(|\vec{v}|\) \(\sqrt{a^2 + b^2}\)
Direction \(\theta\) \(\tan^{-1}\!\left(\frac{b}{a}\right)\), adjusted by quadrant
Unit vector \(\hat{v}\) \(\frac{\vec{v}}{\|\vec{v}\|}\)
Zero vector \(\vec{0}\) \(\langle 0, 0 \rangle\), no direction

Question 1 of 5

Which of the following is a vector quantity?

Question 2 of 5

What is the component form of the vector from \(A(1, -2)\) to \(B(4, 6)\)?

Question 3 of 5

Find the magnitude of \(\vec{v} = \langle -5, 12 \rangle\).

Question 4 of 5

The notation \(\|\vec{v}\|\) represents the vector's:

Question 5 of 5

Two vectors have the same magnitude but point in opposite directions. Are they equal?