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| − | {{ExerciseNavigation|previous=[[Selftest:Minors and cofactors|Minors and cofactors]]|article=[[Matrices]]|next=[[Selftest:Transpose|Transpose]]}} | + | {{ExerciseNavigation|previous=[[Selftest: Minors and cofactors|Minors and cofactors]]|chapter=[[Matrices]]|article=[[Multiplication with a scalar]]|next=[[Selftest: Transpose|Transpose]]}} |
| | + | <br/> |
| | | | |
| − | <quiz> | + | <quiz display=simple> |
| − | {'''Which ...?''' | + | {'''Is there any scalar constant <math>c</math>, so that the following equation holds?'''<br/> |
| − | [[File:Vektorrechnung_Aufgabe6.1.png|200px|left]] | + | :<math> |
| − | <div style="float:left;">
| + | \left[\begin{array}{ccc}1&2&3\\0&2&1\\2&3&0\end{array}\right]\cdot c = \left[\begin{array}{ccc}2&4&6\\0&4&2\\3&6&0\end{array}\right] |
| − | <br style="clear:both;" />
| + | </math><br/> |
| | | typ="()" } | | | typ="()" } |
| − | + <math>\begin{pmatrix} 0 \\ 6 \end{pmatrix}</math>
| + | - <math>c=0.5</math> |
| − | - <math>\begin{pmatrix} 2 \\ 3 \end{pmatrix}</math> | + | - <math>c=1</math> |
| − | - <math>\begin{pmatrix} 5 \\ 0 \end{pmatrix}</math>
| + | - <math>c=2</math> |
| − | - <math>\begin{pmatrix} -2 \\ 3 \end{pmatrix}</math>
| + | + There is no <math>c</math> |
| − | ||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]]
| + | ||<math>c=2</math> would be correct for all the green colored components in the result matrix: <math>\left[\begin{array}{ccc}{\color{Green}2}&{\color{Green}4}&{\color{Green}6}\\{\color{Green}0}&{\color{Green}4}&{\color{Green}2}\\{\color{Red}3}&{\color{Green}6}&{\color{Green}0}\end{array}\right]</math>. For the red component <math>c=1.5</math> would be right. So there is no general <math>c</math>, that holds for all the components. |
| − | </div>
| |
| | | | |
| − | {'''Which of the following vectors forms the sum of <math>\vec{\mathbf{a}}</math> and <math>\vec{\mathbf{b}}</math>?''' | + | {'''Is there any scalar constant <math>c</math>, so that the following equation holds?'''<br/> |
| − | [[File:Vektorrechnung_Aufgabe6.2.png|200px|left]]
| + | :<math> |
| − | <div style="float:left;"> | + | \left[\begin{array}{ccc}2&0&1\\0&2&3\\1&1&0\end{array}\right]\cdot c = \left[\begin{array}{ccc}4&0&2\\0&4&6\\2&2&0\end{array}\right] |
| − | <br style="clear:both;" /> | + | </math><br/> |
| | | typ="()" } | | | typ="()" } |
| − | + <math>\begin{pmatrix} -5 \\ 6 \end{pmatrix}</math>
| + | - <math>c=0.5</math> |
| − | - <math>\begin{pmatrix} -2 \\ 0 \end{pmatrix}</math> | + | - <math>c=1</math> |
| − | - <math>\begin{pmatrix} 0 \\ -3 \end{pmatrix}</math>
| + | + <math>c=2</math> |
| − | - <math>\begin{pmatrix} -6 \\ 5 \end{pmatrix}</math> | + | - There is no <math>c</math> |
| − | ||Vector <math>\vec{\mathbf{a}}</math> only has an x-component, vector <math>\vec{\mathbf{b}}</math> in contrast only has a y-component. The resulting vector consists of the x-component of <math>\vec{\mathbf{a}}</math> and the y-component of <math>\vec{\mathbf{b}}</math>. Further information: see [[Simple arithmetic operations]] | + | ||If all the components of the left matrix are multiplied by <math>c=2</math>, it results in the right matrix. |
| − | </div>
| |
| | | | |
| − | {'''Which of the following vectors forms the sum of <math>\vec{\mathbf{a}}</math> and <math>\vec{\mathbf{b}}</math>?''' | + | {'''Fill in the correct values of the resulting matrix:'''<br/> |
| − | [[File:Vektorrechnung_Aufgabe6.3.png|200px|left]]
| + | |typ="{}" } |
| − | <div style="float:left;">
| + | <math>\left[\begin{array}{ccc}3&2&1\\2&1&2\\2&3&1\end{array}\right]\cdot 3 =</math> |
| − | <br style="clear:both;" /> | + | <br/>{ 9 _2 } { 6 _2 } { 3 _2 }<br/>{ 6 _2 } { 3 _2 } { 6 _2 }<br/>{ 6 _2 } { 9 _2 } { 3 _2 } |
| − | | typ="()" } | |
| − | - <math>\begin{pmatrix} 0 \\ 6 \end{pmatrix}</math>
| |
| − | - <math>\begin{pmatrix} 2 \\ 3 \end{pmatrix}</math>
| |
| − | + <math>\begin{pmatrix} 3 \\ 4 \end{pmatrix}</math>
| |
| − | - <math>\begin{pmatrix} -4 \\ 3 \end{pmatrix}</math>
| |
| − | ||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]]
| |
| − | </div>
| |
| − | | |
| − | | |
| − |
| |
| − | {'''Which of the following vectors forms the substraction <math>\vec{\mathbf{a}}-\vec{\mathbf{b}}</math>?''' | |
| − | [[File:Vektorrechnung_Aufgabe7.1.png|200px|left]]
| |
| − | <div style="float:left;">
| |
| − | <br style="clear:both;" /> | |
| − | | typ="()" }
| |
| − | + <math>\begin{pmatrix} 0 \\ -6 \end{pmatrix}</math>
| |
| − | - <math>\begin{pmatrix} 7 \\ -3 \end{pmatrix}</math>
| |
| − | - <math>\begin{pmatrix} 0 \\ 6 \end{pmatrix}</math>
| |
| − | - <math>\begin{pmatrix} -4,5 \\ 2 \end{pmatrix}</math>
| |
| − | ||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>
| |
| | | | |
| − | {'''Which of the following vectors forms the substraction <math>\vec{\mathbf{a}}-\vec{\mathbf{b}}</math>?''' | + | {'''Is the following equation correct?'''<br/> |
| − | [[File:Vektorrechnung_Aufgabe7.2.png|200px|left]]
| + | :<math>\left[\begin{array}{ccc}2&0&1\\4&1&3\\2&2&1\end{array}\right]\cdot 3 = \left[\begin{array}{ccc}6&0&3\\12&2&9\\6&6&3\end{array}\right]</math><br/> |
| − | <div style="float:left;">
| |
| − | <br style="clear:both;" /> | |
| | | typ="()" } | | | typ="()" } |
| − | - <math>\begin{pmatrix} 3 \\ 0 \end{pmatrix}</math> | + | - Yes |
| − | - <math>\begin{pmatrix} -2 \\ -3 \end{pmatrix}</math>
| + | + No |
| − | + <math>\begin{pmatrix} 4 \\ -2 \end{pmatrix}</math> | + | ||The central component in the resulting matrix is <math>2</math>. But it has to be <math>3</math>. So the equation is not correct. |
| − | - <math>\begin{pmatrix} -4,5 \\ 2 \end{pmatrix}</math>
| |
| − | ||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>
| |
| | | | |
| − | {'''Which of the following vectors forms the substraction <math>\vec{\mathbf{a}}-\vec{\mathbf{b}}</math>?''' | + | {'''Is the following equation correct?'''<br/> |
| − | [[File:Vektorrechnung_Aufgabe7.3.png|200px|left]]
| + | :<math> |
| − | <div style="float:left;">
| + | \left[\begin{array}{ccc}1&0&2\\2&1&3\\3&2&1\end{array}\right]\cdot 2 = \left[\begin{array}{ccc}2&0&4\\4&2&6\\6&4&2\end{array}\right] |
| − | <br style="clear:both;" /> | + | </math><br/> |
| | | typ="()" } | | | typ="()" } |
| − | + <math>\begin{pmatrix} 0 \\ -6 \end{pmatrix}</math> | + | + Yes |
| − | - <math>\begin{pmatrix} 3 \\ -3 \end{pmatrix}</math>
| + | - No |
| − | - <math>\begin{pmatrix} 0 \\ 8 \end{pmatrix}</math> | + | ||The multiplication holds for each of the components. |
| − | - <math>\begin{pmatrix} -2,5 \\ 0,5 \end{pmatrix}</math>
| |
| − | ||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>
| |
| − | | |
| − | {'''Which statement is true?'''}
| |
| − | | |
| − | ''(multiple answers possible)''
| |
| − | | |
| − | - 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>.
| |
| − | - 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>.
| |
| − | + 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>.
| |
| − | ||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]]
| |
| − | | |
| − | | |
| − | | |
| − | {'''Fill-in-the-blank text:'''
| |
| − | | |
| − | Fill in the following words:
| |
| − | | |
| − | ''negative value, same direction, zero vector, factor''
| |
| − | | |
| − | | type="{}" }
| |
| − | 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> |
