4 questions help pls (MATRIX 3D)

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 \(\mathbf{A} \begin{pmatrix} 1 \\ -1 \\ -1 \end{pmatrix}\)