91Ó°ÊÓ

Find the acute angle between the lines. $$ x+2 y=7, \quad 5 x-y=2 $$

Find an equation of the plane. The plane that contains the line \(x=1+t, y=2-t\) \(z=4-3 t\) and is parallel to the plane \(5 x+2 y+z=1\)

If a child pulls a sled through the snow on a level path with a force of \(50 \mathrm{N}\) exerted at an angle of \(38^{\circ}\) above the horizontal, find the horizontal and vertical components of the force.

Reduce the equation to one of the standard forms, classify the surface, and sketch it. $$ y^{2}=x^{2}+\frac{1}{9} z^{2} $$

Describe in words the region of \(\mathbb{R}^{3}\) represented by the equation(s) or inequality. $$ x^{2}+y^{2}=4, \quad z=-1 $$

\(31-32\). 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.) $$ y=x^{2}, \quad y=x^{3} $$

Find an equation of the plane. The plane through the points \((0,1,1),(1,0,1),\) and \((1,1,0)\)

(a) Find a nonzero vector orthogonal to the plane through the points \(P, Q,\) and \(R,\) and \((b)\) find the area of triangle \(P Q R .\) $$ P(0,-2,0), \quad Q(4,1,-2), \quad R(5,3,1) $$

A quarterback throws a football with angle of elevation \(40^{\circ}\) and speed \(60 \mathrm{ft} / \mathrm{s}\). Find the horizontal and vertical components of the velocity vector.

Describe in words the region of \(\mathbb{R}^{3}\) represented by the equation(s) or inequality. $$ x^{2}+y^{2}=4 $$