Vector Algebra

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Problem: Find the decomposition of the vector \( a=(-2 ; 2 ; 2) \) into vectors \( b=(2 ; 1 ; 2) \), \( c=(0 ; 2 ;-2), d=(1 ; 2 ; 0) \).

1.1.1 Vector Algebra

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Problem: 1. Find the dot product of vectors \( a=(\Gamma ; H ; 1 ; \Gamma ; 2) \) and \( b=(3 ; \Gamma ; 2 ; H ; 1) \). 2. Find the angle between vectors \( a=(1 ; H ; \Gamma ; 1 ; 2) \) and \( b=(2 ; H ; \Gamma ; 3 ; 1) \). 3. Find the cross product of vectors \( a=(1 ; H ; \Gamma) \) and \( b=(2 ; H ; \Gamma) \). Where \( \Gamma=2, H=2 \).

1.1.2 Vector Algebra

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Problem: 1. Find the angle \( C \) in the triangle \( A B C \), where \( A=(2 ; \Gamma ; H), B=(3 ; H ; \Gamma), C=(-3 ; 1 ; \Gamma) \). 2. Find the area of the triangle \( A B C \). 3. Find the volume of the tetrahedron \( A B C D \), where \( \quad A=(1 ; \Gamma ; H), \quad B=(-3 ; H ; \Gamma) \), \( C=(\Gamma ; 3 ;-1), D=(2 ; 7 ; \Gamma) \). Where \( \Gamma=2, H=2 \).

1.1.3 Vector Algebra

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Problem: Given vectors \( \vec{a}=\overrightarrow{O A}, \vec{b}=\overrightarrow{O B}, \vec{c}=\overrightarrow{O C}, \vec{d}=\overrightarrow{O D} \). Beams \( O A, O B \) and \( O C \) are edges of e trihedral angle \( T \). 1) Prove that the vectors \( \vec{a}, \vec{b}, \vec{c} \) are linearly independent. 2) Decompose vector \( \vec{d} \) into vectors \( \vec{a}, \vec{b}, \vec{c} \) (solve the resulting system of equations using the inverse matrix). 3) Determine whether point \( D \) lies inside \( T \), outside \( T \), on one of the boundaries of \( T \) (which one?). 4) Determine for what values of the real parameter \( \lambda \) the vector \( \vec{d}+\lambda \vec{a} \), plotted from the point \( O \), lies inside the trihedral angle \( T \). \[ \vec{a}=\overrightarrow{\{3 ; 2 ; 1\}}, \quad \vec{b}=\overrightarrow{\{1 ;-1 ;-2\}}, \quad \vec{c}=\overrightarrow{\{-2 ; 3 ; 5\}}, \quad \vec{d}=\overrightarrow{\{7 ; 4 ; 1} . \]

1.1.4 Vector Algebra

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Problem: Vector \( X=(7 ; 7 ; 2) \) is given in the basis \( e=\left\{e_{1}, e_{2}, e_{3}\right\} \). Find the coordinates of \( X \) in the basis of \( e^{\prime}=\left\{e_{1}^{\prime}, e_{2}^{\prime}, e_{3}^{\prime}\right\} \), where \[ \left\{\begin{array}{l} e_{1}^{\prime}=e_{1}+e_{2}+\frac{6}{7} e_{3} \\ e_{2}^{\prime}=-6 e_{1}-e_{2} \\ e_{3}^{\prime}=-e_{1}+e_{2}+e_{3} \end{array}\right. \]

1.1.5 Vector Algebra

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Problem: Find the coordinates of the vector \( X \) in the basis \( e^{\prime}=\left\{e_{1}^{\prime} ; e_{2}^{\prime} ; e_{3}^{\prime}\right\} \), if \( X=(1 ;-4 ; 8) \) in the basis \( e=\left\{e_{1} ; e_{2} ; e_{3}\right\} \) and \[ \left\{\begin{array}{l} e_{1}^{\prime}=e_{1}+e_{2}-3 e_{3} \\ e_{2}^{\prime}=\frac{3}{4} e_{1}-e_{2} \\ e_{3}^{\prime}=-e_{1}+e_{2}+e_{3} \end{array}\right. \]

1.1.6 Vector Algebra

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Problem: Find the coordinates of the vector \( X \) in the basis \( e^{\prime}=\left\{e_{1}^{\prime} ; e_{2}^{\prime} ; e_{3}^{\prime}\right\} \), if in the basis \( e=\left\{e_{1} ; e_{2} ; e_{3}\right\} \), \( X=(2 ; 6 ;-3) \), and the conversion from \( e \) to \( e^{\prime} \) is: \[ \left\{\begin{array}{l} e_{1}^{\prime}=e_{1}+e_{2}-2 e_{3} \\ e_{2}^{\prime}=\frac{2}{3} e_{1}-e_{2} \\ e_{3}^{\prime}=-e_{1}+e_{2}+e_{3} \end{array}\right. \]

1.1.7 Vector Algebra

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Problem: Reduce the following matrix to a normal form by elementary transformations: \[ \left(\begin{array}{ccc} 3 \lambda^{2}-5 \lambda+2 & 0 & 3 \lambda^{2}-6 \lambda+3 \\ 2 \lambda^{2}-3 \lambda+1 & \lambda-1 & 2 \lambda^{2}-4 \lambda+2 \\ 2 \lambda^{2}-2 \lambda & 0 & 2 \lambda^{2}-4 \lambda+2 \end{array}\right) \]

1.1.8 Vector Algebra

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Problem: Decompose the vector \( x=(-3,5,9,3) \) into the sum of two vectors, one of which lies in the span \( \), and the other is orthogonal to this span, if \( a_{1}=(1,1,1,1), \quad a_{2}=(2,-1,1,1) \), \( a_{3}=(2,-7,-1,-1) \).

1.1.9 Vector Algebra

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Problem: Are the vectors \( \overrightarrow{c_{1}} \) and \( \overrightarrow{c_{2}} \), constructed from the vectors \( \vec{a} \) and \( \vec{b} \) collinear? \[ \begin{array}{l} \vec{a}=\{-2 ; 7 ;-1\}, \quad \vec{b}=\{-3 ; 5 ; 2\}, \\ \overrightarrow{c_{1}}=2 \vec{a}+3 \vec{b}, \quad \overrightarrow{c_{2}}=3 \vec{a}+2 \vec{b} . \end{array} \]

1.1.10 Vector Algebra

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Problem: Find the cosine of the angle between the vectors \( \overrightarrow{A B} \) and \( \overrightarrow{A C} \). \[ A(6 ; 2 ;-3), \quad B(6 ; 3 ;-2), \quad C(7 ; 3 ;-3) \text {. } \]

1.1.11 Vector Algebra

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Problem: Calculate the area of the parallelogram formed by the vectors \( \vec{a} \) and \( \vec{b} \). \[ \begin{array}{l} \vec{a}=7 \vec{p}+\vec{q}, \quad \vec{b}=\vec{p}-3 \vec{g}, \quad|\vec{p}|=3, \\ |\vec{q}|=1, \quad\left(\widehat{\vec{p}, \vec{q})}=\frac{3 \pi}{4} .\right. \end{array} \]

1.1.12 Vector Algebra

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Problem: Find the decomposition of the vector \( \vec{v} \) into the vectors \( \vec{p}, \vec{q} \) and \( \vec{r} \). \[ \begin{array}{l} \vec{v}=\{8 ; 9 ; 4\}, \quad \vec{p}=\{1 ; 0 ; 1\}, \\ \vec{q}=\{0 ;-2 ; 1\}, \quad \vec{r}=\{1 ; 3 ; 0\} . \end{array} \]

1.1.13 Vector Algebra

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Problem: Are the vectors \( \overrightarrow{c_{1}} \) and \( \overrightarrow{c_{2}} \), formed by the vectors \( \vec{a} \) and \( \vec{b} \) collinear? \[ \begin{array}{ll} \vec{a}=\{-1 ; 2 ;-1\}, & \vec{b}=\{2 ;-7 ; 1\}, \\ \overrightarrow{c_{1}}=6 \vec{a}-2 \vec{b}, & \overrightarrow{c_{2}}=\vec{b}-3 \vec{a} . \end{array} \]

1.1.14 Vector Algebra

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Problem: Find the cosine of the angle between the vectors \( \overrightarrow{A B} \) and \( \overrightarrow{A C} \). \[ A(0 ; 2 ;-4), \quad B(8 ; 2 ; 2), \quad C(6 ; 2 ; 4) \text {. } \]

1.1.15 Vector Algebra

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