#$8.$ Given the planes $x-y+2z=14$ and $-2x+2y+z=-3$ do:

$($a$)$ Find parametric equations of their line of intergection,

$($b$)$ Find symmetric equations of the same line,

$($c$)$ Sketch the line in $3$D and label with their coordinates at least two points on it$.$

$($d$)$ Find $cos()$ where $$ is the angle between the two planes.

#$9.$ In the plane, given the vectors $p=($:$-4,1$:$)$ and $q=($:$1,9$:$)$ do:

$($a$)$ find $q-2p,$

$($b$)$ find $|q-2p|,$

$($c$)$ Find two unit vectors perpendicular to $q-2p.($Switch components, put minuses appropriately to get two vectors perpendicular to $q-2p,$ then make them unit vectors.$)$

#$10.$ Given the quadric surface $36x^2=36-9y^2+4z^2$ do :

$($a$)$ rewrite the equation in standard form $($as in section $12\mathrm{.}6),$

$($b$)$ classify the surface $($i$.$e$.$ determine and state what type of surface it is: ellipsoid, or elliptic paraboloid, or a cone, cylinder, hyperboloid of one or two sheets, etc.$)$

$($c$)$ Find an equation of the intersection of this surface with the $xy-$plane. What type of curve is this?

$($d$)$ sketch the surface in $3$D and label, on your graph, at least four points on the surface with all their coordinates $($the $x-$ and $y-$intercepts might be easy$).$

#$11.$ Given the quadric surface $4x^2-16x+18y=-25-9y^2+4z^2-16z$ do :

$($a$)$ complete squares as appropriate and rewrite the equation in either standard form, or close enough to standard form to classify the surface,

$($b$)$ classify the surface,

$($c$)$ Sketch this surface in $3$D$,$

$($d$)$ around which axis is the graph of this surface symmetric? $($This need not be a coordinate axis, you need to find the equation of a line that plays the role of the axis of symmetry. Describe this line with words, e$.$g$.$ going through a certain specified point and parallel to a certain coordinate axis, or write symmetric equations of this line$),$

$($e$)$ around which point is this surface centrally symmetric? $($One is tempted to call this point the "center" of the surface, but I don't think that this terminology is officially used in our book, no name.$)$

#$12.$ In $3$D: $($a$)$ Sketch the cylinder $x^2+y^2=1,($b$)$ sketch the plane $x+2z=3,($c$)$ sketch the ellipse given with equations $x^2+y^2=1,x+2z=3($i$.$e$.$ the intersection of the cylinder from part $($a$)$ with plane from part $($b$)),$

$($d$)$ find suitable parametric equations of the ellipse from part $($c$),$

$($e$)$ Find the tangent line to the ellipse at the point $(x_0,y_0,z_0)=(\frac{-1}{2},\frac{\sqrt[\phantom{2}]{3}}{2},\frac{7}{4}).$ Hint. First find a suitable and convenient value of $t_0($which may depend on your parametrization$).$