91Ó°ÊÓ

Let \(\mathbf{a}=\langle\sqrt{3} / 3, \sqrt{3} / 3, \sqrt{3} / 3\rangle, \mathbf{b}=\langle 1,-1,0\rangle\), and \(\mathbf{c}=\langle-2,-2,1\rangle\). Find the angle between each pair of vectors.

In Problems 7-16, sketch the graph of the given cylindrical or spherical equation. \(\rho=5\)

Name and sketch the graph of each of the following equations in three-space. $$ 9 x^{2}-y^{2}+9 z^{2}-9=0 $$

Find the distance from \((2,3,-1)\) to (a) the \(x y\)-plane, (b) the \(y\)-axis, and (c) the origin.

$$ \lim _{t \rightarrow 0^{-}}\left\langle e^{-1 / t^{2}}, \frac{t}{|t|},|t|\right\rangle $$

Write both the parametric equations and the symmetric equations for the line through the given point parallel to the given vector. $$(-2,2,-2),(7,-6,3)$$

Find the area of the parallelogram with \(\mathbf{a}=2 \mathbf{i}+2 \mathbf{j}-\mathbf{k}\) and \(\mathbf{b}=-\mathbf{i}+\mathbf{j}-4 \mathbf{k}\) as the adjacent sides.

Name and sketch the graph of each of the following equations in three-space. $$ 4 x^{2}+16 y^{2}-32 z=0 $$

For the two-dimensional vectors \(\mathbf{u}\) and \(\mathbf{v}\) in Problems \(9-12\), find the sum \(\mathbf{u}+\mathbf{v}\), the difference \(\mathbf{u}-\mathbf{v}\), and the magnitudes \(\|\mathbf{u}\|\) and \(\|\mathbf{v}\|\). $$ \mathbf{u}=\langle-1,0\rangle, \mathbf{v}=\langle 3,4\rangle $$

Find the symmetric equations of the line of intersection of the given pair of planes. $$4 x+3 y-7 z=1,10 x+6 y-5 z=10$$