A plane is defined by the equation x -7y - 18z = 0
.
a) What is a normal vector to this plane?
b) Explain how you know that this plane passes through the origin.
c) Write the coordinates of three points on this plane.
A plane is defined by the equation 2x - 5y = 0
.
a) What is a normal vector to this plane?
b) Explain how you know that this plane passes through the origin.
c) Write the coordinates of three points on this plane.
A plane is defined by the equation x= 0
.
a) What is a normal vector to this plane?
b) Explain how you know that this plane passes through the origin.
c) Write the coordinates of three points on this plane.
b) A plane has a normal of \vec{n} = (-\displaystyle{\frac{1}{2}}, \displaystyle{\frac{3}{4}}, \displaystyle{\frac{7}{16}})
and passes through the origin. Determine the Cartesian equation of this plane.
A plane is determined by a normal, \vec{n} = (1, 7, 5)
, and contains the point P(-3, 3, 5)
. Determine a Cartesian equation.
Determine the Cartesian equation of the plane that contains the points A(-2, 3, 1), B(3, 4, 5)
, and C(1, 1, 0)
.
The line with vector equation \vec{r} = (2, 0, 1) + t(-4, 5, 5), s\in \mathbb{R}
, lies on the plane \pi
, as does the point P(1, 3, 0)
. Determine the Cartesian equation of \pi.
Determine unit vectors that are normal to each of the following planes:
3x - 4y + 12z - 1 = 0
A plane contains the point A(2, ,2 -1)
and the line \vec{r} = (1, 1, 5) + s(2, 1, 3), s \in \mathb{R}
. Determine the Cartesian equation of this plane.
Determine the Cartesian equation of the plane containing the point (-1, 1, 0)
and perpendicular to the line joining points (1, 2, 1)
and (3, -2, 0)
.
a. Explain the process you would use to determine the angle formed between two intersecting planes.
b. Determine the angle between the planes x + 2y - 3z - 4 =0
and x + 2y - 1=0
.
a) Determine the angle between the planes x + 2y - 3z - 4 = 0
and x + 2y - 1 = 0
.
Determine the Cartesian equation of the plane that passes through the point P(1, 2, 1) and is perpendicular to the line \displaystyle{\frac{x - 3}{-2}} = \displaystyle{\frac{y + 1}{3}} = \displaystyle{\frac{z +4}{1}}
.
What is the value of k
that makes the planes 4x + ky - 2z + 1 = 0
and 2x + 4y - z + 4 = 0
parallel .
4x + ky - 2z + 1 = 0
and 2x + 4y - z + 4 = 0
What is the value of k
that makes these two planes perpendicular?
Can these two planes:
4x + ky - 2z + 1 = 0
and 2x + 4y - z + 4 = 0
ever be coincident? Explain.
Determine the Cartesian equation of the plane that passes through the points (1, 4, 5)
and (3, 2, 1)
and it s perpendicular to the plane 2x - y + z - 1 = 0
.
Determine an equation of the plane that is perpendicular to the plane x + 2y + 4= 0
, contains the origin, and has a normal that makes an angle of 30^{\circ}
with the z-axis.
Determine the equation of the plane that lies between the points (-1, 2, 4)
and (3, 1, -4)
and is equidistant from them.