How do you determine the vector and parametric equations for the plane through point (1, -2,3) and parallel to the xy-plane?

Question

Emma Aldridge · Accepted Answer

# z=3# and # vecr * ( (0), (0), (1) ) = 3 #

Any plane that is parallel to the #xy#-plane is perpendicular the the #z#-axis. ie it has a normal vector given by:

# vec n = ( (0), (0), (1) ) #

Hence the plane has an equation of the form:

# 0x+0y+1z = k => z=k #

we want the plane to pass through #(1,-2,3)# and so
#k=3#

Hence the required plane equation is

# z=3#

For the vector equation we use the form:

# (vecr-vecr_0) * vecn = 0 => vecr * vecn = vecr_0 * vecn #

which gives us:

# vecr * ( (0), (0), (1) ) = ( (1), (-2), (3) ) * ( (0), (0), (1) ) #
# :. vecr * ( (0), (0), (1) ) = 3 #

We can show this on a 3D plot:

Ezra Adams · Answer

undefined To determine the vector and parametric equations for the plane through the point $ (1, -2, 3) $ and parallel to the xy-plane, follow these steps:

1. Vector Equation:
   Since the plane is parallel to the xy-plane, its normal vector will be perpendicular to the xy-plane, which is the z-axis. Thus, the normal vector will be $ \mathbf{n} = \langle 0, 0, 1 \rangle $.
   Now, since the plane passes through the point $ (1, -2, 3) $, we can use this point as a reference.
   Using the point-normal form of the equation of a plane, the vector equation for the plane is:
   $$ \mathbf{r} = \mathbf{r}_0 + t\mathbf{n} $$
   where $ \mathbf{r} $ is any point on the plane, $ \mathbf{r}_0 $ is the given point $ (1, -2, 3) $, $ t $ is a scalar parameter, and $ \mathbf{n} $ is the normal vector.

2. Parametric Equations:
   We can use the vector equation to derive the parametric equations. Let $ x = x_0 + at $, $ y = y_0 + bt $, and $ z = z_0 + ct $, where $ (x_0, y_0, z_0) $ is the given point $ (1, -2, 3) $, and $ (a, b, c) $ are the direction ratios of the parallel vector to the xy-plane, which are $ (0, 0, 1) $.
   Substitute the values:
   $$ x = 1 + 0t $$
   $$ y = -2 + 0t $$
   $$ z = 3 + 1t $$
   Simplifying these equations, we get:
   $$ x = 1 $$
   $$ y = -2 $$
   $$ z = t + 3 $$
   These are the parametric equations for the plane parallel to the xy-plane and passing through the point $ (1, -2, 3) $.