91Ó°ÊÓ

Sketch the surfaces in Exercises \(13-44.\) CYLINDERS $$4 x^{2}+y^{2}=36$$

In Exercises \(1-16,\) give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$z=y^{2}, \quad x=1$$

Find the component form of the vector. The unit vector obtained by rotating the vector \(\langle 1,0\rangle 135^{\circ}\) counterclockwise about the origin

Water main construction \(A\) water main is to be constructed with a 20\(\%\) grade in the north direction and a 10\(\%\) grade in the east direction. Determine the angle \(\theta\) required in the water main for the turn from north to east.

In Exercises \(15-18\) , a. Find the area of the triangle determined by the points \(P, Q\) ,and \(R .\) b. Find a unit vector perpendicular to plane \(P Q R\) . $$ P(2,-2,1), \quad Q(3,-1,2), \quad R(3,-1,1) $$

Express each vector in the form \(\mathbf{v}=v_{1} \mathbf{i}+\) \(v_{2} \mathbf{j}+v_{3} \mathbf{k}.\) \(\overrightarrow{P_{1} P}_{2}\) if \(P_{1}\) is the point \((5,7,-1)\) and \(P_{2}\) is the point \((2,9,-2)\)

In Exercises \(17-24,\) describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities. $$ \text { b. } x \geq 0, \quad y \geq 0, \quad z=0 \quad \text { b. } x \geq 0, \quad y \leq 0, \quad z=0 $$

Sketch the surfaces in Exercises \(13-44.\) ELLIPSOIDS $$9 x^{2}+y^{2}+z^{2}=9$$

In Exercises \(15-18\) , a. Find the area of the triangle determined by the points \(P, Q\) ,and \(R .\) b. Find a unit vector perpendicular to plane \(P Q R\) . $$ P(-2,2,0), \quad Q(0,1,-1), \quad R(-1,2,-2) $$

Express each vector in the form \(\mathbf{v}=v_{1} \mathbf{i}+\) \(v_{2} \mathbf{j}+v_{3} \mathbf{k}.\) \(\overrightarrow{P_{1} P_{2}}\) if \(P_{1}\) is the point \((1,2,0)\) and \(P_{2}\) is the point \((-3,0,5)\)