91Ó°ÊÓ

Find the area of the parallelogram with vertices \(A(-2,1),\) \(B(0,4), C(4,2),\) and \(D(2,-1)\) .

\(21-32=\) Find an equation of the plane. The plane that passes through the point \((6,0,-2)\) and contains the line \(x=4-2 t, y=3+5 t, z=7+4 t\)

Describe in words the region of \(\mathbb{R}^{3}\) represented by theequations or inequalities. \(x=z\)

Ropes 3 \(\mathrm{m}\) and 5 \(\mathrm{m}\) in length are fastened to a holiday deco- ration that is suspended over a town square. The decoration has a mass of 5 \(\mathrm{kg}\) . The ropes, fastened at different heights, make angles of \(52^{\circ}\) and \(40^{\circ}\) with the horizontal. Find the tension in each wire and the magnitude of each tension.

\(28-30=\) Find a vector function that represents the curve of intersection of the two surfaces. The cylinder \(x^{2}+y^{2}=4\) and the surface \(z=x y\)

Reduce the equation to one of the standard forms, classify the surface, and sketch it. \(x^{2}-y^{2}+z^{2}-2 x+2 y+4 z+2=0\)

\(27-28\) Find the acute angles between the curves at their points of intersection. (The angle between two curves is the angle between their tangent lines at the point of intersection.) $$y=\sin x, \quad y=\cos x, \quad 0 \leqslant x \leqslant \pi / 2$$

Find the area of the parallelogram with vertices \(K(1,2,3),\) \(L(1,3,6), M(3,8,6),\) and \(N(3,7,3) .\)

\(29-32\) Find the scalar and vector projections of b onto a. $$\mathbf{a}=\langle- 5,12\rangle, \quad \mathbf{b}=\langle 4,6\rangle$$

Describe in words the region of \(\mathbb{R}^{3}\) represented by theequations or inequalities. \(x^{2}+z^{2} \leqslant 9\)