91Ó°ÊÓ

Find the direction cosines of \(\mathbf{u}\) and demonstrate that the sum of the squares of the direction cosines is \(1 .\) $$ \mathbf{u}=\langle 0,6,-4\rangle $$

Convert the point from rectangular coordinates to spherical coordinates. \((-4,0,0)\)

Use a computer algebra system to graph the surface. (Hint: It may be necessary to solve for \(z\) and acquire two equations to graph the surface.) $$ 4 y=x^{2}+z^{2} $$

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 \(\mathbf{u}\) and \(\mathbf{v}\) as adjacent sides. $$ (2,-3,4),(0,1,2),(-1,2,0) $$

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

Find the magnitude of \(v\). $$ \mathbf{v}=-10 \mathbf{i}+3 \mathbf{j} $$

Use a computer algebra system to graph the surface. (Hint: It may be necessary to solve for \(z\) and acquire two equations to graph the surface.) $$ x^{2}+y^{2}=\left(\frac{2}{z}\right)^{2} $$

Find an equation of the plane passing through the point perpendicular to the given vector or line.\((2,1,2) \quad \mathbf{n}=\mathbf{i}\)

Convert the point from spherical coordinates to rectangular coordinates. \((4, \pi / 6, \pi / 4)\)

In Exercises 35 and 36 , find the coordinates of the midpoint of the line segment joining the points. \((5,-9,7),(-2,3,3)\)