91Ó°ÊÓ

In Problems 29-34, determine whether each system is overdetermined or underdetermined; then solve each system. $$ \begin{array}{r} 4 y-3 z=6 \\ 2 y+z=1 \\ y+z=0 \end{array} $$

Let $$ \mathbf{u}=\left[\begin{array}{l} 3 \\ 4 \end{array}\right], \quad \mathbf{v}=\left[\begin{array}{r} 1 \\ -2 \end{array}\right], \quad \text { and } \quad \mathbf{w}=\left[\begin{array}{l} -1 \\ -2 \end{array}\right] $$ Compute \(\mathbf{v}-\frac{1}{2} \mathbf{u}\) and illustrate the result graphically.

A triangle has vertices at coordinates \(P=(0,0), Q=(0,3)\), and \(R=(2,0)\). (a) Use basic trigonometry to compute the lengths of all three sides and the measures of all three angles. (b) Use the results of this section to repeat (a).

Let \(A=\left[\begin{array}{lll}1 & 4 & -2\end{array}\right]\) and \(B=\left[\begin{array}{r}-1 \\ 2 \\ 2\end{array}\right]\) (a) Compute \(A B\). (b) Compute \(B A\).

A triangle has vertices at coordinates \(P=(1,2,3), Q=\) \((1,1,2)\), and \(R=(4,2,2)\) (a) Compute the lengths of all three sides. (b) Compute all three angles in both radians and degrees.

Let $$A=\left[\begin{array}{rr} 2 & 1 \\ 0 & -3 \end{array}\right]$$ Find \(A^{2}, A^{3}\), and \(A^{4}\).

Let $$ \mathbf{u}=\left[\begin{array}{l} 3 \\ 4 \end{array}\right], \quad \mathbf{v}=\left[\begin{array}{r} 1 \\ -2 \end{array}\right], \quad \text { and } \quad \mathbf{w}=\left[\begin{array}{l} -1 \\ -2 \end{array}\right] $$ Compute \(\mathbf{u}+\mathbf{v}+\mathbf{w}\) and illustrate the result graphically.

Suppose that $$A=\left[\begin{array}{rr} 1 & -1 \\ 3 & 0 \\ 5 & 2 \end{array}\right] \text { and } B=\left[\begin{array}{lll} 2 & 3 & 1 \\ 1 & 0 & 0 \end{array}\right]$$ Show that \((A B)^{\prime}=B^{\prime} A^{\prime}\).

In Problems 29-34, determine whether each system is overdetermined or underdetermined; then solve each system. $$ \begin{array}{rr} x+y= & -1 \\ 2 x-y= & 7 \\ x-2 y= & 8 \end{array} $$

A triangle has vertices at coordinates \(P=(2,1,5), Q=\) \((-1,3,7)\), and \(R=(2,-4,1)\) (a) Compute the lengths of all three sides. (b) Compute all three angles in both radians and degrees.