Difference between revisions of "Selftest: Matrix multiplication with a scalar"
From Robotics
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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]]|article=[[Matrices]]|next=[[Selftest:Transpose|Transpose]]}} | ||
+ | |||
+ | <quiz> | ||
+ | {'''Which ...?''' | ||
+ | [[File:Vektorrechnung_Aufgabe6.1.png|200px|left]] | ||
+ | <div style="float:left;"> | ||
+ | <br style="clear:both;" /> | ||
+ | | typ="()" } | ||
+ | + <math>\begin{pmatrix} 0 \\ 6 \end{pmatrix}</math> | ||
+ | - <math>\begin{pmatrix} 2 \\ 3 \end{pmatrix}</math> | ||
+ | - <math>\begin{pmatrix} 5 \\ 0 \end{pmatrix}</math> | ||
+ | - <math>\begin{pmatrix} -2 \\ 3 \end{pmatrix}</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]] | ||
+ | </div> | ||
+ | |||
+ | {'''Which of the following vectors forms the sum of <math>\vec{\mathbf{a}}</math> and <math>\vec{\mathbf{b}}</math>?''' | ||
+ | [[File:Vektorrechnung_Aufgabe6.2.png|200px|left]] | ||
+ | <div style="float:left;"> | ||
+ | <br style="clear:both;" /> | ||
+ | | typ="()" } | ||
+ | + <math>\begin{pmatrix} -5 \\ 6 \end{pmatrix}</math> | ||
+ | - <math>\begin{pmatrix} -2 \\ 0 \end{pmatrix}</math> | ||
+ | - <math>\begin{pmatrix} 0 \\ -3 \end{pmatrix}</math> | ||
+ | - <math>\begin{pmatrix} -6 \\ 5 \end{pmatrix}</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]] | ||
+ | </div> | ||
+ | |||
+ | {'''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|left]] | ||
+ | <div style="float:left;"> | ||
+ | <br style="clear:both;" /> | ||
+ | | 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>?''' | ||
+ | [[File:Vektorrechnung_Aufgabe7.2.png|200px|left]] | ||
+ | <div style="float:left;"> | ||
+ | <br style="clear:both;" /> | ||
+ | | typ="()" } | ||
+ | - <math>\begin{pmatrix} 3 \\ 0 \end{pmatrix}</math> | ||
+ | - <math>\begin{pmatrix} -2 \\ -3 \end{pmatrix}</math> | ||
+ | + <math>\begin{pmatrix} 4 \\ -2 \end{pmatrix}</math> | ||
+ | - <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>?''' | ||
+ | [[File:Vektorrechnung_Aufgabe7.3.png|200px|left]] | ||
+ | <div style="float:left;"> | ||
+ | <br style="clear:both;" /> | ||
+ | | typ="()" } | ||
+ | + <math>\begin{pmatrix} 0 \\ -6 \end{pmatrix}</math> | ||
+ | - <math>\begin{pmatrix} 3 \\ -3 \end{pmatrix}</math> | ||
+ | - <math>\begin{pmatrix} 0 \\ 8 \end{pmatrix}</math> | ||
+ | - <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> |
Revision as of 14:26, 18 June 2014
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