91Ó°ÊÓ

What is the \(x\) -coordinate of any point in the yz-plane?

In Exercises \(9-14,\) find an equation in cylindrical coordinates for the equation given in rectangular coordinates. $$ z=5 $$

Find \(u \times v\) and show that it is orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\). $$ \begin{array}{l} \mathbf{u}=\langle 2,-3,1\rangle \\ \mathbf{v}=\langle 1,-2,1\rangle \end{array} $$

Find a set of parametric equations of the line. The line passes through the point (2,3,4) and is perpendicular to the plane given by \(3 x+2 y-z=6\).

Find the area of the parallelogram that has the given vectors as adjacent sides. Use a computer algebra system or a graphing utility to verify your result. $$ \begin{array}{l} \mathbf{u}=\langle 2,-1,0\rangle \\ \mathbf{v}=\langle-1,2,0\rangle \end{array} $$

Find the area of the triangle with the given vertices. (Hint: \(\frac{1}{2}\|\mathbf{u} \times \mathbf{v}\|\) is the area of the triangle having \(u\) and \(v\) as adjacent sides.) $$ (2,-3,4),(0,1,2),(-1,2,0) $$

Find the direction cosines of \(u\) and demonstrate that the sum of the squares of the direction cosines is 1. $$ \mathbf{u}=\langle a, b, c\rangle $$

Use the triple scalar product to find the volume of the parallelepiped having adjacent edges \(\mathbf{u}\) \(\mathbf{v},\) and \(\mathbf{w}\). $$ \begin{array}{l} \mathbf{u}=\mathbf{i}+\mathbf{j} \\ \mathbf{v}=\mathbf{j}+\mathbf{k} \\ \mathbf{w}=\mathbf{i}+\mathbf{k} \end{array} $$

In Exercises 33-36, complete the square to write the equation of the sphere in standard form. Find the center and radius. \(x^{2}+y^{2}+z^{2}-2 x+6 y+8 z+1=0\)

Cross Product In Exercises 33 and \(34,\) (a) find the coordinates of three points \(P, Q,\) and \(R\) in the plane, and determine the vectors \(\overrightarrow{P Q}\) and \(\overrightarrow{P R} .\) (b) Find \(\overrightarrow{P Q} \times \overline{P R}\). What is the relationship between the components of the cross product and the coefficients of the equation of the plane? Why is this true? $$ 4 x-3 y-6 z=6 $$