91Ó°ÊÓ

Draw the coordinate axes relative to u and \(\mathbf{v}\) and locate \(\mathbf{w}\) $$\mathbf{u}=\left[\begin{array}{r} 1 \\ -1 \end{array}\right], \mathbf{v}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right], \mathbf{w}=2 \mathbf{u}+3 \mathbf{v}$$

Determine whether the angle between u and v is acute, obtuse, or a right angle. $$\mathbf{u}=\left[\begin{array}{r} 2 \\ -1 \\ 1 \end{array}\right], \mathbf{v}=\left[\begin{array}{r} 1 \\ -2 \\ -1 \end{array}\right]$$

Perform the indicated calculations. $$[2,0,3,2] \cdot([3,1,1,2]+[3,3,2,1]) \text { in } \mathbb{Z}_{4}^{4} \text { and in } \mathbb{Z}_{5}^{4}$$

Draw the coordinate axes relative to u and \(\mathbf{v}\) and locate \(\mathbf{w}\) $$\mathbf{u}=\left[\begin{array}{r} -2 \\ 3 \end{array}\right], \mathbf{v}=\left[\begin{array}{l} 2 \\ 1 \end{array}\right], \mathbf{w}=-\mathbf{u}-2 \mathbf{v}$$

Solve the given equation or indicate that there is no solution. $$x+3=2 \text { in } \mathbb{Z}_{5}$$

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

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

Determine whether the angle between u and v is acute, obtuse, or a right angle. $$\mathbf{u}=[0.9,2.1,1.2], \mathbf{v}=[-4.5,2.6,-0.8]$$

Solve the given equation or indicate that there is no solution. $$x+5=1 \text { in } \mathbb{Z}_{6}$$

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 \(\mathbf{v}\) $$\mathbf{u}=\left[\begin{array}{r} 1 \\ -1 \end{array}\right], \mathbf{v}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right], \mathbf{w}=\left[\begin{array}{l} 2 \\ 6 \end{array}\right]$$