Vector Algebra · Lesson 1 of 9

Understanding Vectors in Space

“Learn how magnitude and direction work together to describe vectors and locate points in space.”

Learning Objectives

• Distinguish scalar and vector quantities using physical examples. • Read directed segments and identify their initial point, terminal point and magnitude. • Describe position vectors and connect coordinates with direction cosines. • Recognise the main types of vectors and explain when two vectors are equal.

When a number is not enough

A distance of 40 km tells us how far something has travelled, but it does not tell us where it has ended up. A displacement of 40 km towards the north supplies both a size and a direction. Vectors are a language for handling quantities for which direction is part of the information.

Definition

Scalar quantity

A quantity described by a magnitude alone. Examples include time, mass, temperature, distance, speed, area, volume and work.

Definition

Vector quantity

A quantity with both magnitude and direction, represented mathematically by a directed line segment. Displacement, velocity, acceleration, force, weight and momentum are examples.

The distinction depends on the quantity, rather than on whether a compass direction is written beside its value. A force of 10 N is still a vector quantity even when its direction has not been supplied. Speed is scalar, while velocity includes the direction of motion. Likewise, distance travelled and displacement answer different questions.

Example — Classifying quantities

Problem
Classify 5 seconds, 1000 cubic centimetres, a force of 10 N, a speed of 30 km/h, a density of 10 grams per cubic centimetre, and a velocity of 20 m/s north.

1.Time, volume, speed and density are scalars: a direction is not part of their definitions.
2.Force and velocity are vectors. The force is a vector even though the statement does not specify its direction.
3.The units alone do not decide the classification; identify what physical quantity is being measured.

A vector as a directed segment

A straight line has two possible directions. Adding an arrow chooses one of them; restricting that directed line to two endpoints also fixes a length. We write the resulting vector as $A B$ , meaning the displacement from $A$ to $B$ .

Definition

Initial point, terminal point and magnitude

In $A B$ , $A$ is the initial point and $B$ is the terminal point. Its magnitude $∣ A B ∣$ is the nonnegative length of the segment; the arrow indicates its direction.

Reading a directed segment— Read the arrow from its tail A to its head B. Reversing it changes the vector.

The symbols $a$ and $A B$ can name the same vector. The magnitude may be written as $∣ a ∣$ or simply $a$ when that convention has been stated. A negative magnitude is impossible: a minus sign attached to a vector reverses direction rather than making its length negative.

Example — Drawing a compass displacement

Problem
Represent a displacement of 40 km, 30 degrees west of south.

1.Draw north–south and east–west reference directions through a starting point O.
2.Begin at south and turn 30 degrees towards west. This is different from beginning at west and turning towards south.
3.Choose a convenient scale, such as 1 cm for 10 km, and draw a 4 cm arrow in that direction. The arrow represents the displacement, while the scale records its magnitude.

A compass displacement— The marked angle is measured from south towards west, not from west towards south.

Position and direction in three dimensions

Choose an origin $O$ and three mutually perpendicular axes. A point $P (x, y, z)$ has the position vector $r = O P$ . Its endpoint records the point, while its direction and length describe the displacement from the chosen origin.

Magnitude of a position vectorLaTeX

∣ r ∣ = x^{2} + y^{2} + z^{2}

The numbers

x, y, z

are the coordinates of

P

. This is the distance from

O

P

For a nonzero vector, let $α, β, γ$ be its angles with the positive coordinate axes. These are called direction angles. Their cosines, written $l, m, n$ , describe the direction without depending on the vector length.

Direction cosinesLaTeX

l = cos α = \frac{x}{r}, m = cos β = \frac{y}{r}, n = cos γ = \frac{z}{r}

Here

r = ∣ r ∣ > 0

. Negative coordinates give negative corresponding direction cosines.

To see the first relation, drop a perpendicular from $P$ to the x-axis. In the resulting right triangle, the signed axial component is $r cos α = x$ . The same reasoning applies to the other axes. Squaring the three coordinate ratios and adding uses $r^{2} = x^{2} + y^{2} + z^{2}$ .

Constraint on direction cosinesLaTeX

l^{2} + m^{2} + n^{2} = 1

This follows from

(x^{2} + y^{2} + z^{2}) / r^{2} = 1

; it is not a rule that

l + m + n = 1

Direction and an axial projection— The inset right triangle shows why the x-component equals r cos α; the other axes work in the same way.

Numbers proportional to the direction cosines are direction ratios. The coordinates $x, y, z$ of this position vector provide one such triple. Later we will calculate with these triples in component form. To preserve the orientation of a directed vector, the sign of the proportionality factor must also be considered; reversing all three signs reverses direction.

Example — Reading a direction from a point

Problem
Find the magnitude and direction cosines of $O P$ for $P (2, - 3, 6)$ .

1.The magnitude is $r = 2^{2} + (- 3)^{2} + 6^{2} = 49 = 7$ .
2.Divide each signed coordinate by 7: $(l, m, n) = (2/7, - 3/7, 6/7)$ .
3.The check is $l^{2} + m^{2} + n^{2} = (4 + 9 + 36) /49 = 1$ . The negative middle value means the direction angle with the positive y-axis is obtuse.

Recognising different types of vectors

Different names highlight different features of a vector or of a group of vectors. A pair can satisfy more than one description: equal vectors are also collinear, while collinear vectors need not be equal. Keep the length, direction and starting point as three separate pieces of information.

Type	Meaning
Zero or null vector	The endpoints coincide and its magnitude is zero. Write $0$ . It has no unique direction.
Unit vector	Its magnitude is one. It represents a direction on a unit scale.
Coinitial vectors	They have the same initial point. Their lengths and directions may differ.
Collinear vectors	They are parallel to the same line; their directions may agree or be opposite.
Equal vectors	They have the same magnitude and the same direction, wherever they are placed.
Negative vectors	$- a$ has the same magnitude as $a$ but the opposite direction.

The chapter uses free vectors. This means that an arrow can be translated parallel to itself without changing the vector it represents. A position vector, however, is drawn from the chosen origin when it is being used to locate a particular point.

Example — Comparing arrows on a square

Problem
Let $A, B, C, D$ be consecutive vertices of a square. Compare $A B$ with $D C$ , $C D$ and $A D$ .

1. $A B$ and $D C$ have equal lengths and the same direction, so they are equal free vectors.
2. $A B$ and $C D$ are collinear and have equal magnitudes, but their directions are opposite: $C D = - A B$ .
3. $A B$ and $A D$ are coinitial and perpendicular. Equal lengths alone do not make them equal vectors.

Equal and coinitial vectors on a square— AB and DC are equal vectors. AB and AD share their initial point.

Common mistake

Equal magnitude does not imply equal vectors. Collinearity also does not imply equality. The zero vector has no unique direction, so do not assign it direction angles or direction cosines.

Quick check

Which quantity is a vector?

Quick check

If $A B$ is reversed, what happens?

Quick check

Two arrows have equal lengths but point in opposite directions. They are

Quick check

Which relation is true for direction cosines?

Quick check

What prevents assigning direction cosines to the zero vector?

Practice Problems

Classify mass, electric field, distance, acceleration, resistance and momentum as scalars or vectors. Explain two of your decisions.
Draw a displacement of 40 km, 30 degrees east of north, using a stated scale.
For $P (- 2, 1, 2)$ , find $∣ O P ∣$ and its direction cosines.
Draw three coinitial vectors, of which two are collinear but unequal.
Give a counterexample to each claim: equal magnitudes imply collinearity; collinearity and equal magnitudes imply equality.
Explain the difference between $AA$ and a unit vector. Can either have negative magnitude?

Key Takeaways

A vector combines magnitude and direction; a scalar has no directional information. Arrow order matters: reversing the endpoints reverses the vector. Position vectors locate points relative to an origin; direction cosines describe nonzero-vector direction. Equal, collinear and coinitial describe different relationships.

Start of chapter

Next · Lesson 2

Adding and Scaling Vectors

When a number is not enough

A vector as a directed segment

Position and direction in three dimensions

Recognising different types of vectors

Quiz

Practice Problems

Key Takeaways