Difference between revisions of "Selftest: Dot product"

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{{ExerciseNavigation|previous=[[Selftest:Simple arithmetic operations|Simple arithmetic operations]]|article=[[Vector algebra]]|next=[[Selftest:Cross product|Cross product]]}}
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{{ExerciseNavigation|previous=[[Selftest: Simple arithmetic operations|Simple arithmetic operations]]|chapter=[[Vector algebra]]|article=[[Dot product]]|next=[[Selftest: Cross product|Cross product]]}}
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<quiz display=simple>
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{'''What is the result of dor product of <math>\vec{\mathbf{a}}</math> and <math>\vec{\mathbf{b}}</math>?'''
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[[File: Vektorrechnung_aufgabe10.1.png|300px|left]]
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<br style="clear:both;" />
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| typ="()" }
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- <math>\begin{pmatrix} 3 \\  5 \end{pmatrix}</math>
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- 1,5
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+ 15
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||The result of the dot product is a ''scalar'' value. In this case the easiest way to compute the dot product is using the angle between the two vectors: <math>\vec{\mathbf{a}} \cdot \vec{\mathbf{b}} =  a b \cos \alpha</math>. The angle is 0° and so the cosine is 1. Further information: see [[Dot product]].
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{'''What is the result of dor product of <math>\vec{\mathbf{a}}</math> and <math>\vec{\mathbf{b}}</math>?'''
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[[File: Vektorrechnung_aufgabe10.2.png|300px|left]]
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<br style="clear:both;" />
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| typ="()" }
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- <math>\begin{pmatrix} 5 \\  3 \end{pmatrix}</math>
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- 15
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+  0
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||The result of the dot product is a ''scalar'' value. In this case the easiest way to compute the dot product is using the angle between the two vectors: <math>\vec{\mathbf{a}} \cdot \vec{\mathbf{b}} =  a b \cos \alpha</math>. The angle is 90° and so the cosine is 0. Further information: see [[Dot product]].
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{'''What is the result of dor product of <math>\vec{\mathbf{a}}</math> and <math>\vec{\mathbf{b}}</math>?'''
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[[File: Vektorrechnung_aufgabe10.3.png|300px|left]]
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<br style="clear:both;" />
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| typ="()" }
 +
- <math>\begin{pmatrix} 5 \\  3 \end{pmatrix}</math>
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+ -15
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- 15
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||The result of the dot product is a ''scalar'' value. In this case the easiest way to compute the dot product is using the angle between the two vectors: <math>\vec{\mathbf{a}} \cdot \vec{\mathbf{b}} =  a b \cos \alpha</math>. The angle is 180° and so the cosine is -1. Further information: see [[Dot product]].
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{'''Please solve the following exercise:'''
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| type="{}" }
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<math>\begin{pmatrix} 2 \\ 3  \\ 5 \end{pmatrix}\cdot \begin{pmatrix} -3 \\ 6  \\ 4 \end{pmatrix}=</math>{ 32 }
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||There are two possibilities to compute the dot product. Either the component representation <math>\vec{\mathbf{a}}\cdot\vec{\mathbf{b}} = \begin{pmatrix}a_1 \\ a_2 \\ a_3 \end{pmatrix}\cdot\begin{pmatrix}b_1 \\ b_2 \\ b_3 \end{pmatrix}=a_1b_1+a_2b_2+a_3b_3</math> is used or the angle between the vectors: <math>\vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a \vec{\mathbf{e}}_{a} \cdot \vec{\mathbf{e}}_{b} = a b \cos \alpha </math>.
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{'''Please solve the following exercise:'''
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| type="{}" }
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<math>\begin{pmatrix} 1 \\ 2  \\ 3 \end{pmatrix}\cdot \begin{pmatrix} -4 \\ 1  \\ -6 \end{pmatrix}=</math>{ -20 }
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||There are two possibilities to compute the dot product. Either the component representation <math>\vec{\mathbf{a}}\cdot\vec{\mathbf{b}} = \begin{pmatrix}a_1 \\ a_2 \\ a_3 \end{pmatrix}\cdot\begin{pmatrix}b_1 \\ b_2 \\ b_3 \end{pmatrix}=a_1b_1+a_2b_2+a_3b_3</math> is used or the angle between the vectors: <math>\vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a \vec{\mathbf{e}}_{a} \cdot \vec{\mathbf{e}}_{b} = a b \cos \alpha </math>.
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 +
 
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{'''Please solve the following exercise:'''
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| type="{}" }
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<math>\begin{pmatrix} 0 \\ 1  \\ 0 \end{pmatrix}\cdot \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}=</math> { 1 }
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||There are two possibilities to compute the dot product. Either the component representation <math>\vec{\mathbf{a}}\cdot\vec{\mathbf{b}} = \begin{pmatrix}a_1 \\ a_2 \\ a_3 \end{pmatrix}\cdot\begin{pmatrix}b_1 \\ b_2 \\ b_3 \end{pmatrix}=a_1b_1+a_2b_2+a_3b_3</math> is used or the angle between the vectors: <math>\vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a \vec{\mathbf{e}}_{a} \cdot \vec{\mathbf{e}}_{b} = a b \cos \alpha </math>.
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</quiz>
  
 
[[Category:Selftest]]
 
[[Category:Selftest]]
 
[[Category:Vectors]]
 
[[Category:Vectors]]

Latest revision as of 10:18, 25 September 2014

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

1. What is the result of dor product of \vec{\mathbf{a}} and \vec{\mathbf{b}}?

Vektorrechnung aufgabe10.1.png


\begin{pmatrix} 3 \\ 5 \end{pmatrix}
1,5
15
→The result of the dot product is a scalar value. In this case the easiest way to compute the dot product is using the angle between the two vectors: \vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a b \cos \alpha. The angle is 0° and so the cosine is 1. Further information: see Dot product.

2. What is the result of dor product of \vec{\mathbf{a}} and \vec{\mathbf{b}}?

Vektorrechnung aufgabe10.2.png


\begin{pmatrix} 5 \\ 3 \end{pmatrix}
15
0
→The result of the dot product is a scalar value. In this case the easiest way to compute the dot product is using the angle between the two vectors: \vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a b \cos \alpha. The angle is 90° and so the cosine is 0. Further information: see Dot product.

3. What is the result of dor product of \vec{\mathbf{a}} and \vec{\mathbf{b}}?

Vektorrechnung aufgabe10.3.png


\begin{pmatrix} 5 \\ 3 \end{pmatrix}
-15
15
→The result of the dot product is a scalar value. In this case the easiest way to compute the dot product is using the angle between the two vectors: \vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a b \cos \alpha. The angle is 180° and so the cosine is -1. Further information: see Dot product.

4. Please solve the following exercise:

\begin{pmatrix} 2 \\ 3 \\ 5 \end{pmatrix}\cdot \begin{pmatrix} -3 \\ 6 \\ 4 \end{pmatrix}=
→ There are two possibilities to compute the dot product. Either the component representation \vec{\mathbf{a}}\cdot\vec{\mathbf{b}} = \begin{pmatrix}a_1 \\ a_2 \\ a_3 \end{pmatrix}\cdot\begin{pmatrix}b_1 \\ b_2 \\ b_3 \end{pmatrix}=a_1b_1+a_2b_2+a_3b_3 is used or the angle between the vectors: \vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a \vec{\mathbf{e}}_{a} \cdot \vec{\mathbf{e}}_{b} = a b \cos \alpha .

5. Please solve the following exercise:

\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}\cdot \begin{pmatrix} -4 \\ 1 \\ -6 \end{pmatrix}=
→ There are two possibilities to compute the dot product. Either the component representation \vec{\mathbf{a}}\cdot\vec{\mathbf{b}} = \begin{pmatrix}a_1 \\ a_2 \\ a_3 \end{pmatrix}\cdot\begin{pmatrix}b_1 \\ b_2 \\ b_3 \end{pmatrix}=a_1b_1+a_2b_2+a_3b_3 is used or the angle between the vectors: \vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a \vec{\mathbf{e}}_{a} \cdot \vec{\mathbf{e}}_{b} = a b \cos \alpha .

6. Please solve the following exercise:

\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\cdot \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}=
→ There are two possibilities to compute the dot product. Either the component representation \vec{\mathbf{a}}\cdot\vec{\mathbf{b}} = \begin{pmatrix}a_1 \\ a_2 \\ a_3 \end{pmatrix}\cdot\begin{pmatrix}b_1 \\ b_2 \\ b_3 \end{pmatrix}=a_1b_1+a_2b_2+a_3b_3 is used or the angle between the vectors: \vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a \vec{\mathbf{e}}_{a} \cdot \vec{\mathbf{e}}_{b} = a b \cos \alpha .

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