91Ó°ÊÓ

Find parametric equations for the lines. The line through the point (3,-2,1) parallel to the line \(x=1+2 t, y=2-t, z=3 t\)

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

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

Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$x^{2}+y^{2}=4, \quad z=-2$$

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\(_\mathbf{v}\) \(\mathbf{u}\) $$\mathbf{v}=-\mathbf{i}+\mathbf{j}, \quad \mathbf{u}=\sqrt{2} \mathbf{i}+\sqrt{3} \mathbf{j}+2 \mathbf{k}$$

Find the length and direction (when defined) of \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u}.\) $$\mathbf{u}=-8 \mathbf{i}-2 \mathbf{j}-4 \mathbf{k}, \quad \mathbf{v}=2 \mathbf{i}+2 \mathbf{j}+\mathbf{k}$$

Match the equation with the surface it defines. Also, identify each surface by type (paraboloid, ellipsoid, etc.). The surfaces are labeled (a)-(1). $$x^{2}+2 z^{2}=8$$

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. $$\frac{3}{5} \mathbf{u}+\frac{4}{5} \mathbf{v}$$

Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$x^{2}+z^{2}=4, \quad y=0$$

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\(_\mathbf{v}\) \(\mathbf{u}\) $$\mathbf{v}=5 \mathbf{i}+\mathbf{j}, \quad \mathbf{u}=2 \mathbf{i}+\sqrt{17} \mathbf{j}$$