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1) Find a direction vector

\(\mathbf{d} = \begin{pmatrix}d_1 \\ d_2 \\d_3 \end{pmatrix}\)

for the line through  \(B = (1, 1, 2)\) and C= (2, 3, 1) such that \(d_1 + d_2 + d_3 = 10\)

 

2) Let u,v and w be vectors satisfying

\(\mathbf{u}\bullet \mathbf{v} = 3, \mathbf{u} \bullet \mathbf{w} = 4, \mathbf{v} \bullet \mathbf{w} = 5.\)
Then what ae

\(\mathbf{u} + 2 \mathbf{v})\bullet \mathbf{w}, (\mathbf{w} - \mathbf{u})\bullet \mathbf{v}, (3\mathbf{v} - 2 \mathbf{w})\bullet \mathbf{u}\)
equal to? Enter the list in the order above.

 

3) Consider the vectors \(\mathbf{v} = \begin{pmatrix} 1\\2\\1 \end{pmatrix}, \mathbf{w} = \begin{pmatrix} 1\\4 \\5 \end{pmatrix}\), and \(\mathbf{x} = \begin{pmatrix}-1 \\ 6 \\ 15\end{pmatrix}\)

If the vectors v.w and x aren't linearly independent, find coefficients a,b  and c, not all 0, such that

\(a\begin{pmatrix} 1\\2\\1 \end{pmatrix}+b \begin{pmatrix} 1\\4 \\5 \end{pmatrix} + c\begin{pmatrix}-1 \\ 6 \\ 15\end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}\)
and answer with \(\dfrac{a-b}{c}\).

 

4)If there exists a matrix A such that

\(\mathbf{A} \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \mathbf{A} \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} -3 \\ 4 \\ 0 \end{pmatrix}, \mathbf{A} \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} -2 \\ 6 \\ 3 \end{pmatrix}\)
calculate \(\mathbf{A} \begin{pmatrix} 1 \\ -1 \\ -1 \end{pmatrix}\)
 

 Mar 10, 2019
 #1
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2.

 

( u + 2v) (dot) w  =

 

u (dot) w   +  2v (dot) w  =

 

4   + 2(v (dot) w)  =

 

4 + 2( 5)  =  10

 

 

(w - u) (dot ) v =

 

w(dot)v   -  u(dot)v =

 

5  -  3     =    2

 

 

(3v - 2w)(dot) u =

 

3v (dot)u   -  2w (dot) u =

 

3 [ v (dot) u ]  - 2 [ w (dot) u ] =

 

3 [3]  - 2[4]  =

 

9  - 8  =   1

 

 

So

 

( u + 2v) (dot) w  ,  (w - u) (dot ) v , (3v - 2w)(dot) u  =   10, 2 , 1

 

 

cool cool cool

 Mar 10, 2019

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