S0.4 Vectors
[A] Definitions
A vector is a quantity which has both magnitude (positive) and direction.
A scalar is a quantity which has magnitude (positive or negative) only.
[B] Notation
Vector status is shown variously by arrows (
) or bold font
(A).
The magnitude of the vector
is denoted by
or
simply A .
A tip!
RevealA tip!
Using vector notation carelessly is probably the most frequent error in introductory physics classes. Although[C] Utility
Vectors allow the laws of physics to be formulated concisely.
[D] The vector sum
- The sum of two vectors
and
is a
vector 
- The magnitude and direction of
are defined
geometrically by the ‘nose-to-tail’ construction. - Distinguish carefully between
and A+B.
[E] Components and unit vectors
- A vector can be represented by a set of components referred to a coordinate system.
- Consider a (two-dimensional: 2D) rectangular coordinate system, defined by two mutually perpendicular (x, y) axes.
Any vector
can be written as

where - Ax and Ay are the x and y components of
(the projections of
on the x and y axes) - î and
are unit vectors (vectors of unit magnitude) along x and y axes.
- Ax and Ay are the x and y components of
- Use of the component-representation allows vector addition ‘by algebra’.
[F] The dot product
Key Point 0.3
The dot product (or scalar product) of two vectors- is written as

- is a scalar
- is given by
where θ is the angle between
and
.
Visualization
The dot product of two vectors reflects the size of the projection (‘shadow’) of one on the other.

Example
The work done by a force
While you will recall the mantra that
WORK = FORCE × DISTANCEyou will probably not have met the more complete and concise vector formulation that follows. It will be introduced properly in S3.2. We give it here so as to show you a scalar product ‘in action’.
The work W done by a force
moving a particle
through a displacement
is
![\[ W = \xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{\vec{F} \cdot \vec{r}} \]](mastermathpng-26.png)
- Equivalently: W=Frcosθ
- In words: work=force × distance moved in direction of force
Dot products of unit vectors:

î⋅î=1×1×cos(0)=1
- Hence the component representation (in 2D):
![\[ \vec{A} \cdot \vec{B} = (A_x\hat{i}+ A_y\hat{j})\cdot (B_x\hat{i}+ B_y\hat{j}) =A_x B_x + A_y B_y \]](mastermathpng-28.png)
[G] Cross product
Key Point 0.4
The cross product (or vector product) of- is written as

- is a vector
- has the magnitude ABsinθ
where θ is the (smaller) angle between
and 
- has direction perpendicular to the plane of
and
in
the sense in which a corkscrew would move if turned so as to take
into
(the corkscrew rule).
Visualization
The cross product of two vectors is (in magnitude) the area of the parallelogram with sides formed from the two vectors.
Example
The torque exerted by a force
You should have an intuitive idea of what is meant by ‘torque’; and you may have met a basic definition along the lines
The torque
exerted (about some point, chosen as origin)
by a force
acting at a point
from the origin is
- magnitude of torque is τ=rFsinθ=Fr⊥
- direction of torque is into the plane
More?
Note that the magnitude of the torque recovers the ‘simple’ non vector-definition: r⊥ is just the ‘perpendicular distance between the line of action of the force and the pivot point’.
The direction assigned to the torque (generally, perpendicular to the plane of the vectors
and
; here into the plane) is
not at all intuitive.
[H] One-dimensional vectors
Key Point 0.5
A one-dimensional (1D) vector is represented by a scalar whose sign (positive/negative) indicates the direction (right/left) along the 1D axis.Learning Resources
| HRW Chapter 3 | |








