MYP Extended Mathematics Topic Vector Representation Revision Notes

MYP Extended Mathematics Topic Vector Representation Revision Notes | RevisionDojo

Vectors Describe Translations and Displacements

Definition

Vector

A quantity with both magnitude (size) and direction, often represented by a directed line segment (an arrow).

A useful way to think about a vector is as something that carries you from one place to another.
In a city laid out on a grid, instructions like "3 blocks east and 4 blocks north" naturally describe a vector.
On a coordinate grid, we often represent a translation using a column vector.

Definition

Column vector

A vector written in component form as $\binom{x}{y}$, representing a translation of $x$ units horizontally and $y$ units vertically.

Vectors are also frequently expressed using the standard basis vectors (or unit vectors) $\mathbf{i}=\binom{1}{0}$ and $\mathbf{j}=\binom{0}{1}$.
In this notation, $\binom{x}{y} = x\mathbf{i} + y\mathbf{j}$.
A column vector $\binom{x}{y}$ means:
1. move $x$ units in the $x$-direction (right if $x>0$, left if $x<0$)
2. move $y$ units in the $y$-direction (up if $y>0$, down if $y<0$)

Two closely related vector ideas appear constantly in geometry:

$\overrightarrow{AB}$ is the vector representing the translation (displacement) from point $A$ to point $B$.
$\overrightarrow{OA}$ is the position vector of point $A$, meaning the vector from the origin $O$ to $A$.

Definition

Position vector

The vector from the origin $O$ to a point, written $\overrightarrow{OA}$.

If $A(x_1,y_1)$ and $B(x_2,y_2)$, then $$\overrightarrow{AB}=\binom{x_2-x_1}{y_2-y_1}$$

Note

The column vector $\binom{0}{0}$ is the zero vector.
It represents "no movement" and has magnitude 0.

Vector Addition Matches Doing One Translation Then Another

When you add vectors, you combine translations.
If you first do translation $\mathbf{a}$ and then do translation $\mathbf{b}$, the overall effect is the single translation $\mathbf{a}+\mathbf{b}$.

Definition

Resultant vector

The single vector that has the same effect as performing several vectors in sequence, for example $\mathbf{a}+\mathbf{b}$.

Pictorial (Geometric) Addition: Head-to-Tail Rule

To draw $\mathbf{a}+\mathbf{b}$:

Draw $\mathbf{a}$.

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(on a website) an arrangement whereby access is restricted to users who have paid to subscribe to the site.

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Note

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Lorem ipsum dolor sit amet, consectetur adipiscing elit.
Sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.
Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris.
Duis aute irure dolor in reprehenderit in voluptate velit esse cillum.

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Vectors Describe Translations and Displacements

Definition

Vector

A quantity with both magnitude (size) and direction, often represented by a directed line segment (an arrow).

A useful way to think about a vector is as something that carries you from one place to another.
In a city laid out on a grid, instructions like "3 blocks east and 4 blocks north" naturally describe a vector.
On a coordinate grid, we often represent a translation using a column vector.

Definition

Column vector

A vector written in component form as $\binom{x}{y}$, representing a translation of $x$ units horizontally and $y$ units vertically.

Vectors are also frequently expressed using the standard basis vectors (or unit vectors) $\mathbf{i}=\binom{1}{0}$ and $\mathbf{j}=\binom{0}{1}$.
In this notation, $\binom{x}{y} = x\mathbf{i} + y\mathbf{j}$.
A column vector $\binom{x}{y}$ means:
1. move $x$ units in the $x$-direction (right if $x>0$, left if $x<0$)
2. move $y$ units in the $y$-direction (up if $y>0$, down if $y<0$)

Two closely related vector ideas appear constantly in geometry:

$\overrightarrow{AB}$ is the vector representing the translation (displacement) from point $A$ to point $B$.
$\overrightarrow{OA}$ is the position vector of point $A$, meaning the vector from the origin $O$ to $A$.

Definition

Position vector

The vector from the origin $O$ to a point, written $\overrightarrow{OA}$.

If $A(x_1,y_1)$ and $B(x_2,y_2)$, then $$\overrightarrow{AB}=\binom{x_2-x_1}{y_2-y_1}$$

Note

The column vector $\binom{0}{0}$ is the zero vector.
It represents "no movement" and has magnitude 0.

Vector Addition Matches Doing One Translation Then Another

When you add vectors, you combine translations.
If you first do translation $\mathbf{a}$ and then do translation $\mathbf{b}$, the overall effect is the single translation $\mathbf{a}+\mathbf{b}$.

Definition

Resultant vector

The single vector that has the same effect as performing several vectors in sequence, for example $\mathbf{a}+\mathbf{b}$.

Pictorial (Geometric) Addition: Head-to-Tail Rule

To draw $\mathbf{a}+\mathbf{b}$:

Draw $\mathbf{a}$.

Definition

Paywall

(on a website) an arrangement whereby access is restricted to users who have paid to subscribe to the site.

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Note

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam quis nostrud exercitation.

Excepteur sint occaecat cupidatat non proident

Tip

Lorem ipsum dolor sit amet, consectetur adipiscing elit.
Sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.
Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris.
Duis aute irure dolor in reprehenderit in voluptate velit esse cillum.

Advanced Number and Bounds3 subtopics

Inequalities and Modelling1 subtopic

Function Transformations4 subtopics

Advanced Probability1 subtopic

Geometric Transformations1 subtopic

Variation and Standard Deviation3 subtopics

Trigonometric Functions and Rules4 subtopics

Sequences, Rational Expressions and Equations5 subtopics

Covariance and Correlation3 subtopics

Logarithms and Non-Linear Inequalities3 subtopics

Graph Theory and Vectors4 subtopics

MYP Extended Mathematics Vector Representation Notes

Advanced Number and Bounds3 subtopics

Inequalities and Modelling1 subtopic

Function Transformations4 subtopics

Advanced Probability1 subtopic

Geometric Transformations1 subtopic

Variation and Standard Deviation3 subtopics

Trigonometric Functions and Rules4 subtopics

Sequences, Rational Expressions and Equations5 subtopics

Covariance and Correlation3 subtopics

Logarithms and Non-Linear Inequalities3 subtopics

Graph Theory and Vectors4 subtopics

Vectors Describe Translations and Displacements

Vector Addition Matches Doing One Translation Then Another

Pictorial (Geometric) Addition: Head-to-Tail Rule

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

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Vectors Describe Translations and Displacements

Vector Addition Matches Doing One Translation Then Another

Pictorial (Geometric) Addition: Head-to-Tail Rule

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account