Difference between revisions of "Selftest: Simple arithmetic operations"

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{{ExerciseNavigation|previous=[[Selftest:Unit vector|Unit vector]]|article=[[Vector algebra]]|next=[[Selftest:Dot product|Dot product]]}}
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{{ExerciseNavigation|previous=[[Selftest: Unit vector|Unit vector]]|chapter=[[Vector algebra]]|article=[[Simple arithmetic operations]]|next=[[Selftest: Dot product|Dot product]]}}
  
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<quiz>
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<quiz display=simple>
 
{'''Which of the following vectors forms the sum of <math>\vec{\mathbf{a}}</math> and <math>\vec{\mathbf{b}}</math>?'''  
 
{'''Which of the following vectors forms the sum of <math>\vec{\mathbf{a}}</math> and <math>\vec{\mathbf{b}}</math>?'''  
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[[File:Vektorrechnung_Aufgabe6.1.png|200px|thumb|left]]
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[[File:Vektorrechnung_Aufgabe6.1.png|200px|left]]
 
<div style="float:left;">
 
<div style="float:left;">
 
<br style="clear:both;" />
 
<br style="clear:both;" />
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{'''Which of the following vectors forms the sum of <math>\vec{\mathbf{a}}</math> and <math>\vec{\mathbf{b}}</math>?'''
 
{'''Which of the following vectors forms the sum of <math>\vec{\mathbf{a}}</math> and <math>\vec{\mathbf{b}}</math>?'''
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[[File:Vektorrechnung_Aufgabe6.2.png|200px|thumb|left]]
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[[File:Vektorrechnung_Aufgabe6.2.png|200px|left]]
 
<div style="float:left;">
 
<div style="float:left;">
 
<br style="clear:both;" />
 
<br style="clear:both;" />
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{'''Which of the following vectors forms the sum of <math>\vec{\mathbf{a}}</math> and <math>\vec{\mathbf{b}}</math>?'''
 
{'''Which of the following vectors forms the sum of <math>\vec{\mathbf{a}}</math> and <math>\vec{\mathbf{b}}</math>?'''
−
[[File:Vektorrechnung_Aufgabe6.3.png|200px|thumb|left]]
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[[File:Vektorrechnung_Aufgabe6.3.png|200px|left]]
 
<div style="float:left;">
 
<div style="float:left;">
 
<br style="clear:both;" />
 
<br style="clear:both;" />
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+ <math>\begin{pmatrix} 3 \\ 4 \end{pmatrix}</math>
 
+ <math>\begin{pmatrix} 3 \\ 4 \end{pmatrix}</math>
 
- <math>\begin{pmatrix} -4 \\ 3 \end{pmatrix}</math>
 
- <math>\begin{pmatrix} -4 \\ 3 \end{pmatrix}</math>
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||The x-component of vector <math>\vec{\mathbf{a}}</math> is directed opposite to the x-component of vector <math>\vec{\mathbf{b}}</math>.  
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||The x-component of vector <math>\vec{\mathbf{a}}</math> is directed opposite to the x-component of vector <math>\vec{\mathbf{b}}</math>. So the x-components are substracted. The y-components are added as usual. Further information: see [[Simple arithmetic operations]]
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Daher zieht man die x-Komponenten an dieser Stelle voneinander ab, die y-Komponenten werden wie gehabt aufaddiert. Weitere Erklärung siehe [[Einfache Rechenoperationen mit Vektoren]]
 
 
</div>
 
</div>
  
  
 
   
 
   
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{'''Welcher der nachstehenden Vektoren bildet die Subtraktion <math>\vec{\mathbf{a}}-\vec{\mathbf{b}}</math>?'''
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{'''Which of the following vectors forms the substraction <math>\vec{\mathbf{a}}-\vec{\mathbf{b}}</math>?'''
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[[File:Vektorrechnung_Aufgabe7.1.png|200px|thumb|left]]
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[[File:Vektorrechnung_Aufgabe7.1.png|200px|left]]
 
<div style="float:left;">
 
<div style="float:left;">
 
<br style="clear:both;" />
 
<br style="clear:both;" />
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- <math>\begin{pmatrix} 0 \\ 6 \end{pmatrix}</math>
 
- <math>\begin{pmatrix} 0 \\ 6 \end{pmatrix}</math>
 
- <math>\begin{pmatrix} -4,5 \\ 2 \end{pmatrix}</math>
 
- <math>\begin{pmatrix} -4,5 \\ 2 \end{pmatrix}</math>
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||Da die x-Komponenten der zu subtrahierenden Vektoren gleich sind, wird die x-Komponente des Differenzvektors Null. Die Länge der y-Komponente ergibt sich durch Subtraktion der jeweiligen y-Komponenten. Weitere Erklärung siehe [[Einfache Rechenoperationen mit Vektoren]]
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||Because the x-components are equal, the resulting x-component is zero. The y-component of the resulting vector is the substraction of the y-components. Further information: see [[Simple arithmetic operations]]
 
</div>
 
</div>
  
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{'''Welcher der nachstehenden Vektoren bildet die Subtraktion <math>\vec{\mathbf{a}}-\vec{\mathbf{b}}</math>?'''
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{'''Which of the following vectors forms the substraction <math>\vec{\mathbf{a}}-\vec{\mathbf{b}}</math>?'''
−
[[File:Vektorrechnung_Aufgabe7.2.png|200px|thumb|left]]
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[[File:Vektorrechnung_Aufgabe7.2.png|200px|left]]
 
<div style="float:left;">
 
<div style="float:left;">
 
<br style="clear:both;" />
 
<br style="clear:both;" />
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+ <math>\begin{pmatrix} 4 \\ -2 \end{pmatrix}</math>
 
+ <math>\begin{pmatrix} 4 \\ -2 \end{pmatrix}</math>
 
- <math>\begin{pmatrix} -4,5 \\ 2 \end{pmatrix}</math>
 
- <math>\begin{pmatrix} -4,5 \\ 2 \end{pmatrix}</math>
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||Die Vektorsubtraktion lässt sich auf die Vektoraddition zurückführen, da <math>\vec{\mathbf{a}}-\vec{\mathbf{b}}=\vec{\mathbf{a}}+(-\vec{\mathbf{b}})</math> gilt. Weitere Erklärung siehe [[Einfache Rechenoperationen mit Vektoren]] [[Bild:Vektorrechnung Vektoraddition und -subtraktion.jpg|300px|thumb|left|Die Vektorsubtraktion]]
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||The substraction of vectors can be traced back to vector addition because <math>\vec{\mathbf{a}}-\vec{\mathbf{b}}=\vec{\mathbf{a}}+(-\vec{\mathbf{b}})</math>. Further information: see [[Simple arithmetic operations]] [[File:Vectoralgebra_addition_substraction.png|300px|left|Vector subtraction]]
 
</div>
 
</div>
  
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{'''Welcher der nachstehenden Vektoren bildet die Subtraktion <math>\vec{\mathbf{a}}-\vec{\mathbf{b}}</math>?'''
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{'''Which of the following vectors forms the substraction <math>\vec{\mathbf{a}}-\vec{\mathbf{b}}</math>?'''
−
[[File:Vektorrechnung_Aufgabe7.3.png|200px|thumb|left]]
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[[File:Vektorrechnung_Aufgabe7.3.png|200px|left]]
 
<div style="float:left;">
 
<div style="float:left;">
 
<br style="clear:both;" />
 
<br style="clear:both;" />
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- <math>\begin{pmatrix} 0 \\ 8 \end{pmatrix}</math>
 
- <math>\begin{pmatrix} 0 \\ 8 \end{pmatrix}</math>
 
- <math>\begin{pmatrix} -2,5 \\ 0,5 \end{pmatrix}</math>
 
- <math>\begin{pmatrix} -2,5 \\ 0,5 \end{pmatrix}</math>
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||Da die x-Komponenten der zu subtrahierenden Vektoren gleich sind, wird die x-Komponente des Differenzvektors Null. Die Länge der y-Komponente ergibt sich durch Subtraktion der jeweiligen y-Komponenten. Weitere Erklärung siehe [[Einfache Rechenoperationen mit Vektoren]]
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||Because the x-components are equal, the resulting x-component is zero. The y-component of the resulting vector is the substraction of the y-components. Further information: see [[Simple arithmetic operations]]
 
</div>
 
</div>
  
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{'''Welche Aussage stimmt?'''}
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{'''Which statement is true?'''}
  
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''(mehrer Antworten sind möglich)''
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''(multiple answers possible)''
  
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- Zur Berechnung des Differenzvektors <math>\vec{\mathbf{a}}-\vec{\mathbf{b}}</math> bildet man zunächst den Vektor <math>-\vec{\mathbf{a}}</math>, indem man bei dem Vektor <math>\vec{\mathbf{a}}</math> die Richtung umkehrt.
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- For the calculation of the difference vector <math>\vec{\mathbf{a}}-\vec{\mathbf{b}}</math> first the vector <math>-\vec{\mathbf{a}}</math> is formed by inverting the direction of <math>\vec{\mathbf{a}}</math>.
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- Zur Berechnung des Differenzvektors <math>\vec{\mathbf{a}}-\vec{\mathbf{b}}</math> bildet man zunächst den Vektor <math>-\vec{\mathbf{b}}</math>, indem man bei dem Vektor <math>\vec{\mathbf{a}}</math> die Richtung umkehrt.
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- For the calculation of the difference vector <math>\vec{\mathbf{a}}-\vec{\mathbf{b}}</math> first the vector <math>-\vec{\mathbf{b}}</math> is formed by inverting the direction of <math>\vec{\mathbf{a}}</math>.
−
+ Zur Berechnung des Differenzvektors <math>\vec{\mathbf{a}}-\vec{\mathbf{b}}</math> bildet man zunächst den  Vektor <math>-\vec{\mathbf{b}}</math>, indem man bei dem Vektor <math>\vec{\mathbf{b}}</math> die Richtung umkehrt.
+
+ For the calculation of the difference vector <math>\vec{\mathbf{a}}-\vec{\mathbf{b}}</math> first the vector <math>-\vec{\mathbf{b}}</math> is formed by inverting the direction of <math>\vec{\mathbf{b}}</math>.
−
||[[Bild:Vektorrechnung Vektoraddition und -subtraktion.jpg|300px|thumb|left|Die Vektorsubtraktion]] Die Vektorsubtraktion lässt sich auf die Vektoraddition zurückführen, da <math>\vec{\mathbf{a}}-\vec{\mathbf{b}}=\vec{\mathbf{a}}+(-\vec{\mathbf{b}})</math> gilt. Weitere Erklärung siehe [[Einfache Rechenoperationen mit Vektoren]]  
+
||The substraction of vectors can be traced back to vector addition because <math>\vec{\mathbf{a}}-\vec{\mathbf{b}}=\vec{\mathbf{a}}+(-\vec{\mathbf{b}})</math>. Further information: see [[Simple arithmetic operations]] [[File:Vectoralgebra_addition_substraction.png|300px|left|Vector substraction]]
  
  
  
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{'''Lückentext:'''
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{'''Fill-in-the-blank text:'''
  
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Fügen Sie folgende Wörter ein und achten Sie dabei auf Groß- und Kleinschreibung:
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Fill in the following words:
  
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''negative Zahl, gleicher Richtung, Nullvektor, Faktor''
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''negative value, same direction, zero vector, factor''
  
 
| type="{}" }
 
| type="{}" }
−
Multipliziert man einen Vektor <math>\vec{\mathbf{a}}</math> mit einer positiven reellen Zahl  ''p'', entsteht ein Vektor<math>\vec{\mathbf{a}}p</math> mit { gleicher Richtung } und verändertem Betrag, der sich um den { Faktor }  <math>{p}</math> geändert hat. Erhält der Vektor <math>\vec{\mathbf{a}}</math> durch die Multiplikation eine entgegengesetzte Richtung, so handelt es sich um eine { negative Zahl }. Für den Sonderfall ''p=0'' erhält man einen { Nullvektor }.
+
Multiplying a vector <math>\vec{\mathbf{a}}</math> by a real value ''p'' results in a vector <math>\vec{\mathbf{a}}_p</math> with { same direction } and different magnitude. The magnitude changes with { factor }  <math>{p}</math>. If the resulting vector <math>\vec{\mathbf{a}}_p</math> has an oppsite direction, ''p'' is a { negative value }. The special case ''p=0'' results in a { zero vector }.
 
</quiz>
 
</quiz>
  

Latest revision as of 10:57, 30 November 2014

← Previous exercise: Unit vector Exercises for chapter Vector algebra | Article: Simple arithmetic operations Next exercise: Dot product →

1. Which of the following vectors forms the sum of \vec{\mathbf{a}} and \vec{\mathbf{b}}?

Vektorrechnung Aufgabe6.1.png


\begin{pmatrix} 0 \\ 6 \end{pmatrix}
\begin{pmatrix} 2 \\ 3 \end{pmatrix}
\begin{pmatrix} 5 \\ 0 \end{pmatrix}
\begin{pmatrix} -2 \\ 3 \end{pmatrix}
→The x-components of the two vectors cancel each other. Therefore the sum vector only has an y-component unequal zero. The length is the sum of the two y-components. Further information: see Simple arithmetic operations

2. Which of the following vectors forms the sum of \vec{\mathbf{a}} and \vec{\mathbf{b}}?

Vektorrechnung Aufgabe6.2.png


\begin{pmatrix} -5 \\ 6 \end{pmatrix}
\begin{pmatrix} -2 \\ 0 \end{pmatrix}
\begin{pmatrix} 0 \\ -3 \end{pmatrix}
\begin{pmatrix} -6 \\ 5 \end{pmatrix}
→Vector \vec{\mathbf{a}} only has an x-component, vector \vec{\mathbf{b}} in contrast only has a y-component. The resulting vector consists of the x-component of \vec{\mathbf{a}} and the y-component of \vec{\mathbf{b}}. Further information: see Simple arithmetic operations

3. Which of the following vectors forms the sum of \vec{\mathbf{a}} and \vec{\mathbf{b}}?

Vektorrechnung Aufgabe6.3.png


\begin{pmatrix} 0 \\ 6 \end{pmatrix}
\begin{pmatrix} 2 \\ 3 \end{pmatrix}
\begin{pmatrix} 3 \\ 4 \end{pmatrix}
\begin{pmatrix} -4 \\ 3 \end{pmatrix}
→The x-component of vector \vec{\mathbf{a}} is directed opposite to the x-component of vector \vec{\mathbf{b}}. So the x-components are substracted. The y-components are added as usual. Further information: see Simple arithmetic operations

4. Which of the following vectors forms the substraction \vec{\mathbf{a}}-\vec{\mathbf{b}}?

Vektorrechnung Aufgabe7.1.png


\begin{pmatrix} 0 \\ -6 \end{pmatrix}
\begin{pmatrix} 7 \\ -3 \end{pmatrix}
\begin{pmatrix} 0 \\ 6 \end{pmatrix}
\begin{pmatrix} -4,5 \\ 2 \end{pmatrix}
→Because the x-components are equal, the resulting x-component is zero. The y-component of the resulting vector is the substraction of the y-components. Further information: see Simple arithmetic operations

5. Which of the following vectors forms the substraction \vec{\mathbf{a}}-\vec{\mathbf{b}}?

Vektorrechnung Aufgabe7.2.png


\begin{pmatrix} 3 \\ 0 \end{pmatrix}
\begin{pmatrix} -2 \\ -3 \end{pmatrix}
\begin{pmatrix} 4 \\ -2 \end{pmatrix}
\begin{pmatrix} -4,5 \\ 2 \end{pmatrix}
→The substraction of vectors can be traced back to vector addition because \vec{\mathbf{a}}-\vec{\mathbf{b}}=\vec{\mathbf{a}}+(-\vec{\mathbf{b}}). Further information: see Simple arithmetic operations
Vector subtraction

6. Which of the following vectors forms the substraction \vec{\mathbf{a}}-\vec{\mathbf{b}}?

Vektorrechnung Aufgabe7.3.png


\begin{pmatrix} 0 \\ -6 \end{pmatrix}
\begin{pmatrix} 3 \\ -3 \end{pmatrix}
\begin{pmatrix} 0 \\ 8 \end{pmatrix}
\begin{pmatrix} -2,5 \\ 0,5 \end{pmatrix}
→Because the x-components are equal, the resulting x-component is zero. The y-component of the resulting vector is the substraction of the y-components. Further information: see Simple arithmetic operations

7. Which statement is true?

For the calculation of the difference vector \vec{\mathbf{a}}-\vec{\mathbf{b}} first the vector -\vec{\mathbf{a}} is formed by inverting the direction of \vec{\mathbf{a}}.
For the calculation of the difference vector \vec{\mathbf{a}}-\vec{\mathbf{b}} first the vector -\vec{\mathbf{b}} is formed by inverting the direction of \vec{\mathbf{a}}.
For the calculation of the difference vector \vec{\mathbf{a}}-\vec{\mathbf{b}} first the vector -\vec{\mathbf{b}} is formed by inverting the direction of \vec{\mathbf{b}}.
→The substraction of vectors can be traced back to vector addition because \vec{\mathbf{a}}-\vec{\mathbf{b}}=\vec{\mathbf{a}}+(-\vec{\mathbf{b}}). Further information: see Simple arithmetic operations
Vector substraction

8. Fill-in-the-blank text:

Fill in the following words:

negative value, same direction, zero vector, factor

Multiplying a vector \vec{\mathbf{a}} by a real value p results in a vector \vec{\mathbf{a}}_p with and different magnitude. The magnitude changes with {p}. If the resulting vector \vec{\mathbf{a}}_p has an oppsite direction, p is a . The special case p=0 results in a .

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