Difference between revisions of "Selftest: Dot product"

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@@ Line 1: / Line 1: @@
 {{ExerciseNavigation|previous=[[Selftest:Simple arithmetic operations|Simple arithmetic operations]]|article=[[Vector algebra]]|next=[[Selftest:Cross product|Cross product]]}}
+<quiz>
+{'''What is the result of dor product of <math>\vec{\mathbf{a}}</math> and <math>\vec{\mathbf{b}}</math>?'''
+[[File: Vektorrechnung_aufgabe10.1.png|300px|left]]
+<br style="clear:both;" />
+| typ="()" }
+- <math>\begin{pmatrix} 3 \\  5 \end{pmatrix}</math>
+- 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:<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]].
+{'''What is the result of dor product of <math>\vec{\mathbf{a}}</math> and <math>\vec{\mathbf{b}}</math>?'''
+[[File: Vektorrechnung_aufgabe10.2.png|300px|left]]
+<br style="clear:both;" />
+| typ="()" }
+- <math>\begin{pmatrix} 5 \\  3 \end{pmatrix}</math>
+- 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:<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]].
+{'''What is the result of dor product of <math>\vec{\mathbf{a}}</math> and <math>\vec{\mathbf{b}}</math>?'''
+[[File: Vektorrechnung_aufgabe10.3.png|300px|left]]
+<br style="clear:both;" />
+| typ="()" }
+- <math>\begin{pmatrix} 5 \\  3 \end{pmatrix}</math>
++ -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:<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]].
+{'''Bitte lösen Sie folgende Aufgabe:'''
+| type="{}" }
+<math>\begin{pmatrix} 2 \\ 3  \\ 5 \end{pmatrix}\cdot \begin{pmatrix} -3 \\ 6  \\ 4 \end{pmatrix}=</math>{ 32 }
+||Es gibt zwei Möglichkeiten zur Berechnung des Skalarprodukts. Entweder berechnet man es mit Hilfe der Komponentendarstellung:<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> oder man nutzt den eingeschlossenen Winkel: <math>\vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a \vec{\mathbf{e}}_{a} \cdot \vec{\mathbf{e}}_{b} = a b \cos \alpha </math>.
+{'''Bitte lösen Sie folgende Aufgabe:'''
+| type="{}" }
+<math>\begin{pmatrix} 1 \\ 2  \\ 3 \end{pmatrix}\cdot \begin{pmatrix} -4 \\ 1  \\ -6 \end{pmatrix}=</math>{ -20 }
+||Es gibt zwei Möglichkeiten zur Berechnung des Skalarprodukts. Entweder berechnet man es mit Hilfe der Komponentendarstellung:<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> oder man nutzt den eingeschlossenen Winkel: <math>\vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a \vec{\mathbf{e}}_{a} \cdot \vec{\mathbf{e}}_{b} = ab \cos \alpha </math>.
+{'''Bitte lösen Sie folgende Aufgabe:'''
+| type="{}" }
+<math>\begin{pmatrix} 0 \\ 1  \\ 0 \end{pmatrix}\cdot \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}=</math> { 1 }
+||Es gibt zwei Möglichkeiten zur Berechnung des Skalarprodukts. Entweder berechnet man es mit Hilfe der Komponentendarstellung:<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> oder man nutzt den eingeschlossenen Winkel: <math>\vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a \vec{\mathbf{e}}_{a} \cdot \vec{\mathbf{e}}_{b} = a b \cos \alpha </math>.
+</quiz>
 [[Category:Selftest]]
 [[Category:Vectors]]

Revision as of 16:26, 23 May 2014

← Previous exercise: Simple arithmetic operations

Exercises for chapter {{{chapter}}} | Article: Vector algebra

Next exercise: Cross product →

Point added for a correct answer:
Points for a wrong answer:
Ignore the questions' coefficients:

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

	$\begin{pmatrix} 3 \\ 5 \end{pmatrix}$
	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 $\vec{\mathbf{b}}$ ?

	$\begin{pmatrix} 5 \\ 3 \end{pmatrix}$
	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 $\vec{\mathbf{b}}$ ?

	$\begin{pmatrix} 5 \\ 3 \end{pmatrix}$
	-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.