Consider the equation of the plane x + 2y + 3z = 12. (a) Find a normal vector to the plane. (b) Find where the x, y and z-axes intersect the plane. Using this information, sketch the first octant portion of the plane. (c) Using the points in part (b), find two non-parallel vectors that are parallel to the plane.
Consider the equation of the plane x + 2y + 3z = 12.
(a) Find a normal vector to the plane.
(b) Find where the x, y and z-axes intersect the plane. Using this information, sketch
the first octant portion of the plane.
(c) Using the points in part (b), find two non-parallel
plane.
(d) Using part (c), find another normal vector to the plane. Show that this vector is
parallel to the vector from part (a).
(e) Using the new normal vector and the dot product definition of the plane, find
an alternative Euclidean equation for the plane. Compare this new equation to
x + 2y + 3z = 12. How are these two equations related? Is it obvious that they
describe the same set of points (x, y, z)?
(f) Using parts (b) and (c), find a vector (parametric) equation that describes the
plane.
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