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.