91Ó°ÊÓ

In Exercises \(9 - 14 ,\) sketch the coordinate axes and then include the vectors \(\mathbf { u } , \mathbf { v } ,\) and \(\mathbf { u } \times \mathbf { v }\) as vectors starting at the origin. $$ \mathbf { u } = \mathbf { i } - \mathbf { k } , \quad \mathbf { v } = \mathbf { j } $$

Find the angles between the vectors to the nearest hundredth of a radian. $$ \mathbf{u}=2 \mathbf{i}-2 \mathbf{j}+\mathbf{k}, \quad \mathbf{v}=3 \mathbf{i}+4 \mathbf{k} $$

Find the angles between the vectors to the nearest hundredth of a radian. $$ \mathbf{u}=\sqrt{3} \mathbf{i}-7 \mathbf{j}, \quad \mathbf{v}=\sqrt{3} \mathbf{i}+\mathbf{j}-2 \mathbf{k} $$

In Exercises \(9 - 14 ,\) sketch the coordinate axes and then include the vectors \(\mathbf { u } , \mathbf { v } ,\) and \(\mathbf { u } \times \mathbf { v }\) as vectors starting at the origin. $$ \mathbf { u } = \mathbf { i } - \mathbf { k } , \quad \mathbf { v } = \mathbf { j } + \mathbf { k } $$

In Exercises \(1-16,\) give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$x^{2}+y^{2}+(z+3)^{2}=25, \quad z=0$$

In Exercises \(9-16,\) find the component form of the vector. The vector from the point \(A=(2,3)\) to the origin

Find parametric equations for the lines. \begin{equation} \quad \text { The }x \text { -axis }\end{equation}

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

In Exercises \(1-16,\) give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$x^{2}+(y-1)^{2}+z^{2}=4, \quad y=0$$

Find parametric equations for the lines. \begin{equation} \quad \text { The }z \text { -axis }\end{equation}