Content code
m1303
Slug (identifier)
adding-and-subtracting-vectors
Parent content
Grades
Secondary V
Topic
Mathematics
Tags
vector
resultant
components
sum
addition of vectors
Chasles Relation
vector subtraction
Content
Contenu
Corps

​Adding or subtracting vectors results in a new vector called the resultant vector (or resultant).

Links
Title (level 2)
Adding Vectors
Title slug (identifier)
adding-vectors
Contenu
Corps

Several methods are used to add vectors. When the vectors are in a plane, the triangle method or parallelogram method can be used. Vectors can also be added algebraically using the algebraic method. Finally, in some cases, Chasles' relation can be used.

Content
Corps

The sum of vectors |\overrightarrow{u}| and |\overrightarrow{v}| is defined by the following resultant vector.||\overrightarrow{u}+\overrightarrow{v}||

Title (level 3)
The Triangle Method
Title slug (identifier)
triangle-method
Corps

As its name suggests, the triangle method consists of obtaining a resultant vector by forming a triangle in a Cartesian plane.

Content
Corps
  1. Place the vectors one after the other so the end of the first vector becomes the origin of the second.

  2. Draw the resultant vector by connecting the origin of the first vector to the end of the second.

Content
Corps

Determine the resultant of the sum of the following vectors.

Image
Two vectors represented in a Cartesian plane.
Columns number
2 columns
Format
50% / 50%
First column
Corps
  1. Place the vectors one after the other

Image
Two vectors placed one after the other according to the triangle method.
Second column
Corps
  1. Draw the resultant vector

Image
Connect the origin of the first vector with the end of the second to obtain the resultant vector.
Corps

The resultant is the green vector, obtained after adding |\overrightarrow {u}| and |\overrightarrow {v}.|

Title (level 3)
The Parallelogram Method
Title slug (identifier)
parallelogram-method
Corps

The parallelogram method can be used when the vectors have the same origin. The resultant is the diagonal of the parallelogram formed.

Content
Corps
  1. Place |\overrightarrow{u}| and |\overrightarrow{v}| so they have the same origin.

  2. Draw another |\overrightarrow{u}| at the end of |\overrightarrow{v}.|

  3. Draw another |\overrightarrow{v}| at the end of |\overrightarrow{u}.|

  4. Draw the resultant vector that starts at the origin and goes to the opposite vertex of the parallelogram formed.

Corps

Content
Corps

Determine the resultant of the sum of the following vectors.

Image
Two vectors represented in a Cartesian plane.
Columns number
2 columns
Format
50% / 50%
First column
Corps
  1. Place |\overrightarrow{u}| and |\overrightarrow{v}| so they have the same origin
    Moving |\overrightarrow{v},| gives the following representation.

Image
Place the two vectors so they have the same origin.
Second column
Corps
  1. Draw another |\overrightarrow{u}| at the end of |\overrightarrow{v}|

Image
Draw another vector u at the end of vector v.
Columns number
2 columns
Format
50% / 50%
First column
Corps
  1. Draw another |\overrightarrow{v}| at the end of |\overrightarrow{u}|

Image
Draw another vector v at the end of the first vector u.
Second column
Corps
  1. Draw the resultant vector

Image
Connect the origin to the opposite vertex of the parallelogram to obtain the resultant vector.
Corps

The resultant is the green vector, obtained after adding |\overrightarrow {u}| and |\overrightarrow {v}.|

Title (level 3)
The Algebraic Method
Title slug (identifier)
algebraic-method-1
Corps

​To add vectors algebraically, add the components of the vectors to obtain the components of the resultant vector.

Content
Corps

Let |\overrightarrow{u}=(a,b)| and |\overrightarrow{v}=(c,d),| therefore:||\begin{align}\overrightarrow {u}+\overrightarrow{v}&= (\color{#EC0000}a,\color{#3B87CD}b)+(\color{#EC0000}c, \color{#3B87CD}d)\\&=(\color{#EC0000}a+\color{#EC0000}c,\color{#3B87CD}b+ \color{#3B87CD}d)\end{align}||

Content
Corps
  1. Add the components in |x| and in |y| of each vector.

  2. Determine the components of the resultant vector.

Content
Corps

Let |\overrightarrow{u}=(3, 2)| and |\overrightarrow{v}=(4, -1).| What is the resultant of the sum of the two vectors?

  1. Add the components in |x| and |y| of each vector||\begin{align}\overrightarrow {u}+\overrightarrow{v}&= (\color{#EC0000}3,\color{#3B87CD}2)+(\color{#EC0000}4, \color{#3B87CD}{-1})\\&=(\color{#EC0000}3+\color{#EC0000}4,\color{#3B87CD}2+ \color{#3B87CD}{-1})\\&=(7,1)\end{align}||

  2. Determine the components of the resultant vector
    Components |(x,y)| of the resultant are |(7,1).|

Title (level 3)
Chasles' Relation
Title slug (identifier)
chasles-relation-1
Corps

To add vectors using Chasles’ relation, the starting and ending points of the vectors must be identified by letters. Chasles' relation is useful during vector demonstrations.

Content
Corps

Two vectors can be added using Chasles’ relation when the extremity of the first vector matches the origin of the second. The sum is then equal to the vector with the origin of the first vector and the extremity of the second.||\overrightarrow{A\color{#FF55C3}{B}}+\overrightarrow{\color{#FF55C3}BC}=\overrightarrow{AC}||

Corps

The following example demonstrates an instance when Chasles’ relation can be applied.

Content
Corps

What is the vector resulting from adding vectors |\overrightarrow{RS}| and |\overrightarrow{TR}?|

Note that point |R| is the first vector’s origin and the second vector’s extremity. Vector addition is commutative, therefore, the order of vector addition can be changed, causing the endpoint of the first to match the origin of the second.||\overrightarrow{RS}+\overrightarrow{TR}=\overrightarrow{TR}+\overrightarrow{RS}|| We now have the correct form to apply Chasles’ relation. We obtain:||\overrightarrow{T\color{#FF55C3}{R}}+\overrightarrow{\color{#FF55C3}RS}=\overrightarrow{TS}|| The resultant of the sum of the vectors |\overrightarrow{RS}| and |\overrightarrow{TR}| is vector |\overrightarrow{TS}.|

Title (level 2)
Subtracting Vectors
Title slug (identifier)
subtracting-vectors
Contenu
Corps

Subtracting a vector is equivalent to adding its opposite vector.​

Content
Corps

Subtracting |\overrightarrow{u}| and |\overrightarrow{v}| results in:||\overrightarrow{u}-\overrightarrow{v}=\overrightarrow{u}+(-\overrightarrow{v})|| This image illustrates adding and subtracting |\overrightarrow{u}| and |\overrightarrow{v}.|

Image
Adding and subtracting two vectors.
Corps

After changing subtraction to addition, the triangle or parallelogram method can be used to obtain the resultant vector in a plane.

Title (level 3)
The Algebraic Method
Title slug (identifier)
algebraic-method-2
Corps

To subtract vectors algebraically, add the components of the first vector with the opposite of the components of the second vector.

Content
Corps

Let |\overrightarrow{u}=(a,b)| and |\overrightarrow{v}=(c,d),| therefore:||\begin{align}\overrightarrow {u}-\overrightarrow{v}&=\overrightarrow {u}+(-\overrightarrow{v})\\&= (\color{#EC0000}a,\color{#3B87CD}b)+(-\color{#EC0000}c, -\color{#3B87CD}d)\\&=(\color{#EC0000}a-\color{#EC0000}c,\color{#3B87CD}b-\color{#3B87CD}d)\end{align}||

Content
Corps
  1. Add the components of the first vector with the opposite of the components of the second vector.

  2. Determine the components of the resultant vector.

Content
Corps

Let vector |\overrightarrow{u}=(3,2)| and vector |\overrightarrow{v}=(4,-1).| What's the value of |\overrightarrow{u}-\overrightarrow{v}?|

  1. Add the components of the first vector with the opposite of the components of the second vector||\begin{align}\overrightarrow {u}-\overrightarrow{v}&=(\color{#EC0000}a,\color{#3B87CD}b)+(-\color{#EC0000}c,-\color{#3B87CD}d)\\&=(\color{#EC0000}3,\color{#3B87CD}2)+(-\color{#EC0000}4,-\color{#3B87CD}{-1})\\&=(\color{#EC0000}3-\color{#EC0000}4,\color{#3B87CD}2+\color{#3B87CD}{1})\\&=(-1,3)\end{align}||

  2. Determine the components of the resultant vector
    Components |(x,y)| of the resultant are |(7, 1).|

Title (level 3)
Chasles' Relation
Title slug (identifier)
chasles-relation-2
Corps

Using the starting and ending point of the vectors, and the definition of vector subtraction, the subtraction is written as follows.||\begin{align}\overrightarrow{AB}-\overrightarrow{CD}&=\overrightarrow{AB}+(-\overrightarrow{CD})\\&=\overrightarrow{AB}+\overrightarrow{DC}\end{align}|| The last example makes it possible to use Chasles’ relation in specific cases.

Content
Corps

What is the vector resulting from the subtraction |\overrightarrow{AC}-\overrightarrow{BC}?|

Since subtracting is the same as adding the opposite vector, the result is:||\overrightarrow{AC}-\overrightarrow{BC}=\overrightarrow{AC}+\overrightarrow{CB}|| As the extremity of the first vector equals the origin of the second, we can apply Chasles’ relation.||\overrightarrow{A\color{#FF55C3}{C}}+\overrightarrow{\color{#FF55C3}CB}=\overrightarrow{AB}||The resultant of subtracting vectors |\overrightarrow{AC}| and |\overrightarrow{BC}| is vector |\overrightarrow{AB}.|

Title (level 2)
See also
Title slug (identifier)
see-also
Contenu
Corps

To validate your understanding about vectors interactively, consult the following CrashLesson:

MiniRécup
Links
Remove audio playback
No