91Ó°ÊÓ

Find parametric equations for the lines. The line through (1,1,1) parallel to the \(z\) -axis

Let \(\mathbf{u}=\langle 3,-2\rangle\) and \(\mathbf{v}=\langle-2,5\rangle .\) Find the \((\mathbf{a})\) component form and (b) magnitude (length) of the vector. $$-\frac{5}{13} \mathbf{u}+\frac{12}{13} \mathbf{v}$$

Find a. \(\mathbf{v} \cdot \mathbf{u},|\mathbf{v}|,|\mathbf{u}|\) b. the cosine of the angle between \(\mathbf{v}\) and \(\mathbf{u}\) c. the scalar component of \(\mathbf{u}\) in the direction of \(\mathbf{v}\) d. the vector proj\(_\mathbf{v}\) \(\mathbf{u}\) $$\mathbf{v}=\left\langle\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{3}}\right\rangle, \quad \mathbf{u}=\left\langle\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{3}}\right\rangle$$

Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$y^{2}+z^{2}=1, \quad x=0$$

Find the length and direction (when defined) of \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u}.\) $$\mathbf{u}=\frac{3}{2} \mathbf{i}-\frac{1}{2} \mathbf{j}+\mathbf{k}, \quad \mathbf{v}=\mathbf{i}+\mathbf{j}+2 \mathbf{k}$$

Find parametric equations for the lines. The line through (2,4,5) perpendicular to the plane \(3 x+7 y-5 z=21\)

Find parametric equations for the lines. The line through (0,-7,0) perpendicular to the plane \(x+2 y+2 z=13\)

Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$x^{2}+y^{2}+z^{2}=1, \quad x=0$$

Find the angles between the vectors to the nearest hundredth of a radian. $$\mathbf{u}=2 \mathbf{i}+\mathbf{j}, \quad \mathbf{v}=\mathbf{i}+2 \mathbf{j}-\mathbf{k}$$

Match the equation with the surface it defines. Also, identify each surface by type (paraboloid, ellipsoid, etc.). The surfaces are labeled (a)-(1). $$x=z^{2}-y^{2}$$