Introduction

In this educational notebook, designed for classroom instruction, we delve into the fundamental concepts of vectors, a crucial topic in mathematics and physics education . Through a series of interactive visualizations and practical examples, students will gain a deep understanding of vectors, their properties, and operations .
The notebook begins by introducing vectors as mathematical quantities with both magnitude and direction . Visual representations and real - world applications are employed to illustrate these concepts, enhancing students' intuitive understanding . The algebraic representation of vectors by coordinates is explored, emphasizing the importance of direction and magnitude .
It provides interactive exercises where students can manipulate vectors graphically, allowing them to grasp vector addition, subtraction, and scalar multiplication intuitively . Engaging visuals and step - by - step explanations make complex vector operations accessible, ensuring students can apply their knowledge effectively . Furthermore, the notebook includes demonstrations of vector addition and subtraction using Mathematica, a powerful computational tool . Through visually appealing plots and clear annotations, students can witness the graphical representation of vector operations, reinforcing their understanding and providing a bridge between theory and application .
It is tailored for high school students and educators, aiming to enhance classroom learning experiences . Attendees of the workshop in December will have the opportunity to explore the notebook' s content firsthand, gaining valuable insights into effective teaching strategies for vectors . By combining theoretical concepts with interactive visualizations, this educational resource aims to inspire and empower both students and educators in the fascinating world of vectors .

Unraveling Magnitude through Coordinates

Vectors represent mathematical quantities with both magnitude (length) and direction . Imagine a vector as an arrow drawn from a starting point (
P
) to an ending point (
Q
), with an arrowhead at (
Q
) indicating its terminal position. Vectors are considered equivalent if they share the same length and direction, irrespective of their initial points.
In essence, vectors provide a way to represent quantities with both size and orientation . This concept is vital in various fields, including physics, engineering, and computer science, where understanding the magnitude and direction of quantities like force, velocity, and displacement is crucial for solving real - world problems . By visualizing vectors as arrows, we can intuitively grasp their properties, making them a fundamental concept in mathematics and its applications .
This concept naturally leads to the algebraic representation of vectors using coordinates. To achieve this, we translate a vector
v
parallel to itself preserving its direction, until its starting point coincides with the origin . Rhe coordinates of
v
then denoted as
(a,b)
, representing the terminal point (
R
) of the translated vector. The l magnitude
||v||
of
v
is simply the length of the line segment connecting the origin to point (
R
) .This approach allows us to represent vectors numerically, making them applicable in various mathematical and real-world contexts.”
||v||
2
a
+
2
b
Here, the double - vertical bars—notation of the next edition of the text—appear, to make it easy to distinguish magnitude of a vector from the absolute value of a real number .

2 - Dimensional Vectors

Suppose we want to find the magnitude of vectors
v
1
and
v
2
.
We express the vectors in Mathematica as
In[]:=
v1={-2,5};​​v2={3,-1};
We know that the magnitude of a vector is the square root of the sum of squares of its components
In[]:=
magnitude1=Sqrt[{-2}^2+5^2]
Out[]=
{
29
}
In[]:=
magnitude2=Sqrt[3^2+{-1}^2]
Out[]=
{
10
}
Mathematica has a built in function, Norm, that computes the magnitude of vectors
In[]:=
Norm[v1]
Out[]=
29
In[]:=
Norm[v2]
Out[]=
10

Vector Operations and their Plots

◼
  • The sum of two vectors
    v(a,b)
    and
    w(c,d)
    is the vector
    v+w(a+c,b+d)
    .
    • In[]:=
    v1+v2
    Out[]=
    {1,4}
    In[]:=
    Show[Graphics[{Red,Arrow[{{0,0},v1}],Text["v1",v1/2,{0,-1}],Blue,Arrow[{{0,0},v2}],Text["v2",v2/2,{0,-1}],Green,Arrow[{{0,0},v1+v2}],Text["v1 + v2",(v1+v2)/2,{0,-1}],Dashing[{0.03,0.02}],Gray,Line[{v1,v1+v2}],Line[{v2,v1+v2}]}],Axes->True,AxesOrigin->{0,0},AxesLabel->{x,y},AxesStyle->Arrowheads[0.03],AspectRatio->Automatic,PlotRange->All,PlotLabel->"Vector Difference",Ticks->{Automatic,Automatic}]
    Out[]=
    ◼
  • The vector difference of
    v(a,b)
    and
    w(c,d)
    in that order is
    v-w(a-c,b-d)
    .
    • In[]:=
    v1-v2
    Out[]=
    {-5,6}
    In[]:=
    Show[Graphics[{Red,Arrow[{{0,0},v1}],Text["v1",v1/2,{0,-1}],Blue,Arrow[{{0,0},-v2}],Text["-v2",-v2/2,{0,-1}],Green,Arrow[{{0,0},v1-v2}],Text["v1 - v2",(v1-v2)/2,{0,-1}],Dashing[{0.03,0.02}],Gray,Line[{v1,v1-v2}],Line[{-v2,v1-v2}]}],Axes->True,AxesOrigin->{0,0},AxesLabel->{x,y},AxesStyle->Arrowheads[0.03],AspectRatio->Automatic,PlotRange->All,PlotLabel->"Vector Difference",Ticks->{Automatic,Automatic}]
    Out[]=
    ◼
  • The product of the vector
    v(a,b)
    by the scalar (real number)
    r
    is
    rv(ra,rb)
    .
    • In[]:=
    5*v1
    Out[]=
    {-10,25}

    Example

    Findandplot
    2
    v
    1
    -3
    v
    2
    andcompute
    ||2
    v
    1
    -3
    v
    2
    ||
    In[]:=
    r=2;
    In[]:=
    r*v1
    Out[]=
    {-4,10}
    In[]:=
    s=3;
    In[]:=
    s*v2
    Out[]=
    {9,-3}
    and
    In[]:=
    w=r*v1-s*v2
    Out[]=
    {-13,13}
    In[]:=
    Norm[w]
    Out[]=
    13
    2
    In[]:=
    Show[Graphics[{Red,Arrow[{{0,0},v1}],Text["v1",v1/2,{0,-1}],Blue,Arrow[{{0,0},v2}],Text["v2",v2/2,{0,-1}],Green,Arrow[{{0,0},w}],Text["w",(w)/2,{0,-1}]}],Axes->True,AxesOrigin->{0,0},AxesLabel->{x,y},AxesStyle->Arrowheads[0.03],AspectRatio->Automatic,PlotRange->All,PlotLabel->"Vector Difference",Ticks->{Automatic,Automatic}]
    Out[]=

    3 - Dimensional Vectors

    Vector products

    Dot product

    The dot product of any two vectors of the same dimension can be done with the dot operation given as Dot[vector 1, vector 2]
    Consider the vectors:

    Cross product

    The cross product can be done on two vectors. It is important to note that the cross product is an operation that is only functional in three dimensions. The operation can be computed using the Cross[vector 1, vector 2] operation

    Vector Angles

    The definition of a dot product yields
    We can compute the angle between our two vectors :

    Example

    Now, consider the following vectors

    Addition

    Subtraction

    Linear combination of the vectors

    References

    1
    .
    James F. Hurley, 2002, University of Connecticut, Department of Mathematics, 196 Auditorium Road Unit 3009, Storrs CT 06269-3009. https://www2.math.uconn.edu/~hurley/math220/Mathematica_docs/VectPlot.pdf
    2
    .
    http://dslavsk.sites.luc.edu/courses/phys301/classnotes/mVectorIntro.pdf
    3
    .
    MATHEMATICA TUTORIAL Part 2.1: Vectors
    ​https://www.cfm.brown.edu/people/dobrush/am34/Mathematica/ch1/vector.html

    CITE THIS NOTEBOOK

    Geometric and Algebraic Representations of Vectors​
    by Athanasios Paraskevopoulos​
    Wolfram Community, STAFF PICKS, October 10, 2023
    ​https://community.wolfram.com/groups/-/m/t/3030841