Review Problems for Quiz 1 (9.1-10.3)

  1. Given four points in an XYZ coordinates: A(0,0,0), B(1, -1, 0), C(0, 1, 1), and D(-1, 0, 1):

    2.  
    1. Find the distance between A and D.
    2. Find triple product of AB, BC, and AB.
    3. Find the area of the parallelogram formed by the vectors AB and AC. Are A, B, and C are in a same line?
    4. Find a unit vector which is orthogonal to both AB and AC.
    5. Find the angle between AB and AC.
    6. Find the equation of the sphere which centered at D with the radius 2.
    7. Find the equation of line which passes through A and B in both parametric and symmetric form.
    8. Find the mid-point  between A and B.
    9. Are A, B, C,  and D co-planar? If not, find the volume of the parallelepiped formed from AB, AC, AD.
    10. Find the equation of plane which passes through C and orthogonal to the vector AB.
    11. Find the distance from the point D to the plane above.

      12.  
  2. Name the following equations (line, plane, surface, paraboloid, ... ). It the equation is a line, find the direction of the line and a point on the line; if it is a plane, find the normal direction and a point on the plane; if it is a sphere, find the radius and the center; if it is a quadric surface, sketch it.

    3.  
    1. 2 x = 3 z + 5
    2. 2x + 5 = y, and x + y + z =0
    3. (x-5)/3  =  y/0  = (z +1)/(-1)
    4. x2 + y2 + z2 + 4 x - 6 y + 2 z + 6 = 0
    5. x = y2 + z2
    6. y2 = x2 + z2
    7. f(x,y,z) = 0,  or y = g(x,z)
    8. f1(x,y,z) = 0 and f2(x,y,z)=0.

      9.  
  3. Given r(t) = (2 t ,  3 sint , -3 cos t):

    4.  
  4. Write down the parametric equation for the plane curve  y = sin x  in XYZ coordinates.
  5. Write down the parametric equation of a plane circle: x2 + y2 = R2 in XYZ-space and find the curvature of the circle.

    6.