91Ó°ÊÓ

For the following exercises, lines \(L_{1}\) and \(L_{2}\) are given. Determine whether the lines are equal, parallel but not equal, skew, or intersecting. \(L_{1} : x=y-1=-z\) and \(L_{2} : x-2=-y=\frac{z}{2}\)

For the following exercises, lines \(L_{1}\) and \(L_{2}\) are given. Determine whether the lines are equal, parallel but not equal, skew, or intersecting. \(L_{1} : x=2 t, y=0, z=3, \quad t \in \mathbb{R}\) and \(L_{2} : x=0, y=8+s, z=7+s, \quad s \in \mathbb{R}\)

For the following exercises, lines \(L_{1}\) and \(L_{2}\) are given. Determine whether the lines are equal, parallel but not equal, skew, or intersecting. \(L_{1} : x=-1+2 t, y=1+3 t, z=7 t, \quad t \in \mathbb{R}\) and \(L_{2} : x-1=\frac{2}{3}(y-4)=\frac{2}{7} z-2\)

For the following exercises, lines \(L_{1}\) and \(L_{2}\) are given. Determine whether the lines are equal, parallel but not equal, skew, or intersecting. \(L_{1} : 3 x=y+1=2 z\) and \(L_{2} : x=6+2 t, y=17+6 t, z=9+3 t, \quad t \in \mathbb{R}\)

Consider line \(L\) of symmetric equations \(x-2=-y=\frac{z}{2}\) and point \(A(1,1,1) .\) a. Find parametric equations for a line parallel to L that passes through point A. b. Find symmetric equations of a line skew to L and that passes through point A. c. Find symmetric equations of a line that intersects L and passes through point A.

Consider line \(L\) of parametric equations \(x=t, y=2 t, z=3, \quad t \in \mathbb{R}\) a. Find parametric equations for a line parallel to L that passes through the origin. b. Find parametric equations of a line skew to L that passes through the origin. c. Find symmetric equations of a line that intersects L and passes through the origin.

For the following exercises, point \(P\) and vector \(\mathbf{n}\) are given. a. Find the scalar equation of the plane that passes through \(P\) and has normal vector \(\mathbf{n} .\) b. Find the general form of the equation of the plane that passes through \(P\) and has normal vector \(\mathbf{n} .\) \(P(0,0,0), \quad \mathbf{n}=3 \mathbf{i}-2 \mathbf{j}+4 \mathbf{k}\)

For the following exercises, point \(P\) and vector \(\mathbf{n}\) are given. a. Find the scalar equation of the plane that passes through \(P\) and has normal vector \(\mathbf{n} .\) b. Find the general form of the equation of the plane that passes through \(P\) and has normal vector \(\mathbf{n} .\) \(P(3,2,2), \quad \mathbf{n}=2 \mathbf{i}+3 \mathbf{j}-\mathbf{k}\)

For the following exercises, point \(P\) and vector \(\mathbf{n}\) are given. a. Find the scalar equation of the plane that passes through \(P\) and has normal vector \(\mathbf{n} .\) b. Find the general form of the equation of the plane that passes through \(P\) and has normal vector \(\mathbf{n} .\) \(P(1,2,3), \quad \mathbf{n}=\langle 1,2,3\rangle\)

For the following exercises, point \(P\) and vector \(\mathbf{n}\) are given. a. Find the scalar equation of the plane that passes through \(P\) and has normal vector \(\mathbf{n} .\) b. Find the general form of the equation of the plane that passes through \(P\) and has normal vector \(\mathbf{n} .\) \(P(0,0,0), \quad \mathbf{n}=\langle- 3,2,-1\rangle\)