Review Sheet for Test

  1. Given two plane equations: P1: $3x+4y+7z+4=0$ and P2: $x-y+2z+3=0.$

    1. Find the line equation which is the intersection of these plane and passes through the point $(1,4,5).$

    2. Find the plane equation which is perpendicular to P1 and P2 and passes through $(1,4,5).$

    3. Use Maple to plot your answers for a) and b) above.

  2. Given three points: $A=(1,2,3),$ $B=(0,2,2)$ and $C(-1,3,4).$

    1. Find the plane equation which passes through $A,B,$ and $C.$

    2. Find the area of the triangle $ABC.$

  3. Given four points: $A=(1,2,3),$ $B=(0,2,2)$, $C(-1,3,4),$ and $D=(1,-1,1).$

    1. Determine if these four points lie on the same plane.

    2. If these four points do not lie on the same plane, find the volume of the parallelepiped formed by $A,B,C,$ and $D.$

  4. Discusses (i) the traces (when $x=k,y=k$ and $z=k$ respectively) and (ii) the surfaces for the following 'three dimensional' equations:

    1. MATH

    2. $z=x^{2}-y^{2}.$

    3. $x^{2}+y^{2}=z$

    4. $z=(y-3)^{2}$

    5. $y=(x-3)^{2}$

  5. Converting $x^{2}+y^{2}=2z$ to equations in cylindrical and spherical coordinate systems respectively.

  6. Describe $z=2r$ and write it in rectangular coordinate system.

  7. Find the spherical equation for the sphere which is center at $(0,0,1)$ and has radius $2$.

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