91Ó°ÊÓ

Draw the standard coordinate axes on the same diagram as the axes relative to u and v. Use these to find \(\mathbf{w}\) as a linear combination of u and \(\mathbf{v}\) $$\mathbf{u}=\left[\begin{array}{r} 1 \\ -1 \end{array}\right], \mathbf{v}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right], \mathbf{w}=\left[\begin{array}{l} 2 \\ 6 \end{array}\right]$$

Solve the given equation or indicate that there is no solution. $$2 x=1 \text { in } \mathbb{Z}_{3}$$

Find the vector form of the equation of the line in \(\mathbb{R}^{3}\) that passes through \(P=(-1,0,3)\) and is perpendicular to the plane with general equation \(x-3 y+2 z=5\)

Determine whether the angle between u and v is acute, obtuse, or a right angle. $$\mathbf{u}=[1,-2,3,4], \mathbf{v}=[-3,1,-1,1]$$

Draw the standard coordinate axes on the same diagram as the axes relative to u and v. Use these to find \(\mathbf{w}\) as a linear combination of u and \(\mathbf{v}\) $$\mathbf{u}=\left[\begin{array}{r} -2 \\ 3 \end{array}\right], \mathbf{v}=\left[\begin{array}{l} 2 \\ 1 \end{array}\right], \mathbf{w}=\left[\begin{array}{l} 2 \\ 9 \end{array}\right]$$

Determine whether the angle between u and v is acute, obtuse, or a right angle. $$\mathbf{u}=[1,2,3,4], \mathbf{v}=[5,6,7,8]$$

Solve the given equation or indicate that there is no solution. $$2 x=1 \text { in } \mathbb{Z}_{4}$$

Find the vector form of the equation of the line in \(\mathbb{R}^{3}\) that passes through \(P=(-1,0,3)\) and is parallel to the line with parametric equations \\[ \begin{array}{l} x=1-t \\ y=2+3 t \\ z=-2-t \end{array} \\]

Solve the given equation or indicate that there is no solution. $$2 x=1 \text { in } \mathbb{Z}_{5}$$

Find the normal form of the equation of the plane that passes through \(P=(0,-2,5)\) and is parallel to the plane with general equation \(6 x-y+2 z=3\)