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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 ($\vec{A}$) or bold font (A).

The magnitude of the vector $\vec{A}$ is denoted by $\mid \vec{A} \mid$ or simply A .

A tip!

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A tip!

Using vector notation carelessly is probably the most frequent error in introductory physics classes. Although $\vec{A}$ and A look almost the same they mean different things and obey different rules. If you feel that the mathematics is being asked to convey too much here remember that all languages (mathematics is a ‘language’) do this kind of thing: think of ‘practice’ and ‘practise’ which also ‘obey different rules’!
 

Examples

Examples

positionvector  distancescalar
velocityvector  speedscalar
accelerationvector  energyscalar
 
IA Note on Vector Notation

[C] Utility

Vectors allow the laws of physics to be formulated concisely.

IVectors are useful!

[D] The vector sum

MAddition of Vectors

[E] Components and unit vectors

AComponents of a vector

Example

Example

\begin{eqnarray*} \vec{A} & = & \hat{i} + 2 \hat{j} \\ \vec{B} & = & 3\hat{i} + \hat{j} \\ \vec{C} & = & 4\hat{i} + 3\hat{j} \end{eqnarray*}
Graphic - No title - compvecs
 

[F] The dot product

Key Point 0.3

The dot product (or scalar product) of two vectors $\vec{A}$ and $\vec{B}$:

[G] Cross product

Key Point 0.4

The cross product (or vector product) of $\vec{A}$ with $\vec{B}$:
AThe second-most important use of a corkscrew

Graphic - No title - crossarea

Example

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

 TORQUE = FORCE × PERPENDICULAR DISTANCE FROM AXIS 
The vector formulation that follows will probably be new to you. It will be introduced properly in S5.6. We give it here so as to show you a vector product ‘in action’.

 

Graphic - No title - crosstorque

The torque $\vec{\tau}$ exerted (about some point, chosen as origin) by a force $\vec{F}$ acting at a point $\vec{r}$ from the origin is

\[ \vec{\tau} = \xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{\vec{r} \times \vec{F}} \]

  • magnitude of torque is τ=rFsinθ=Fr
  • direction of torque is into the plane
 

More?

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 $\vec{F}$ and $\vec{r}$; 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.

Example

Example

Graphic - No title - onedvecs
  • Consider two particles moving towards one another on a horizontal track
  • Each has speed v0 (speed is always a positive quantity)
  • Particle 1 is moving to the right: its velocity is v1=+v0
  • Particle 2 is moving to the left: its velocity is v2=-v0
 
TClock Face: The Maximum Dot Product
TClock Face: Cross Product Questions
TVectors Aren't Scalars(!)

Learning Resources

Textbook: HRW Chapter 3
Course Questions:
Self-Test Questions: