91Ó°ÊÓ

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 \(9 - 14 ,\) 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. $$ \mathbf { u } = 2 \mathbf { i } - \mathbf { j } , \quad \mathbf { v } = \mathbf { i } + 2 \mathbf { j } $$

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. $$ \mathbf { u } = \mathbf { i } + \mathbf { j } , \quad \mathbf { v } = \mathbf { i } - \mathbf { j } $$

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

Sketch the surfaces in Exercises \(13-44\) . CYLINDERS $$ x^{2}+y^{2}=4 $$

In Exercises \(9-16,\) find the component form of the vector. The unit vector that makes an angle \(\theta=2 \pi / 3\) with the positive \(x\) -axis

Triangle Find the measures of the angles of the triangle whose vertices are \(A=(-1,0), B=(2,1),\) and \(C=(1,-2) .\)

Find parametrizations for the line segments joining the points. Draw coordinate axes and sketch each segment, indicating the direction of increasing \(t\) for your parametrization. \begin{equation}(0,0,0), \quad(1,1,3 / 2)\end{equation}

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. $$ \mathbf { u } = \mathbf { j } + 2 \mathbf { k } , \quad \mathbf { v } = \mathbf { i } $$

Find parametrizations for the line segments joining the points. Draw coordinate axes and sketch each segment, indicating the direction of increasing \(t\) for your parametrization. \begin{equation}(0,0,0), \quad(1,0,0)\end{equation}