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Geometry and Planes Practice Problems

— McGraw-Hill Professional
Updated on Oct 3, 2011

Geometry and Planes Practice Problems

If necessary, review:

PROBLEM 1

Draw a graph of the plane W represented by the following equation:

−2 x − 4 y + 3 z − 12 = 0

SOLUTION 1

The x -intercept, or the point where the plane W intersects the x axis, can be found by setting y = 0 and z = 0 and solving the resulting equation for x . Call this point P :

Vectors and Cartesian Three-Space Planes Plotting A Plane

 

Therefore,

P = (−6,0,0)

The y -intercept, or the point where the plane W intersects the y axis, can be found by setting x = 0 and z = 0 and solving the resulting equation for y . Call this point Q :

Vectors and Cartesian Three-Space Planes Plotting A Plane

Therefore,

Q = (0, − 3,0)

The z -intercept, or the point where the plane W intersects the z axis, can be found by setting x = 0 and y = 0 and solving the resulting equation for z . Call this point R :

Vectors and Cartesian Three-Space Planes Plotting A Plane

Therefore,

R = (0,0,4)

These three points are shown in the plot of Fig. 9-11. The plane can be envisioned, based on this data. (The dashed axes are “behind” the plane.)

PROBLEM 2

Suppose a plane contains the point (2,−7,0) and a normal vector to the plane at this point is 3 i + 3 j + 2 k . What is the equation of this plane?

SOLUTION 2

The vector 3 i + 3 j + 2 k is equivalent to ( a,b,c ) = (3,3,2). We have one point ( x 0 , y 0 , z 0 ) = (2,−7,0). Plugging these values into the general formula for the equation of a plane gives us the following:

 

 

Vectors and Cartesian Three-Space Planes Plotting A Plane

Fig. 9-11 . Illustration for Problem 9-7. Dashed portions of the coordinate axes are “behind” the plane.

Vectors and Cartesian Three-Space Planes Plotting A Plane

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