Plane Problems

1Determine the equations of the coordinate axes and the coordinate planes.

2Determine the equation of the plane that contains the lines:

Line Contained in a PlaneLine Contained in a Plain

3Determine the equation of the plane that contains the point A = (2, 5, 1) and the line:

Line Contained in a Plane

4Find the intersecting point between the plane x + 2y − z − 2 = 0, the line determined by the point (1, −3, 2) and the vector Vector Contained in a Plain.

5Determine, in intercept form, the equation of the plane that passes through the points A = (2, 0, 0), B = (0, 4, 0) and C = (0, 0, 7).

6π is a plane that passes through P = (1, 2, 1) and intersects the positive coordinate semi-axes at points A, B and C. If ABC is an equilateral triangle, determine the equations of π.

7Find the equation of the plane that passes through the point P = (1, 1, 1) and is parallel to:

Parametric Equation

8Determine the equation of the plane that contains the line Line Equation and is parallel to the line Parametric Equation.

9Calculate the equation of the plane that passes through the point (1, 1, 2) and is parallel to the following lines:

Line Equations


1

Determine the equations of the coordinate axes and the coordinate planes.

X-Axis

Axis Equation

Y-Axis

Axis Equation

Z-Axis

Axis Equation

X-Y Plane

Plane Equation

X-Z Plane

Plane Equation

Y-Z Equation

Plane Equation


2

Determine the equation of the plane that contains the lines:

Line Equation Line Equation

Vectors and Point

Cartesian Equation of the Plane


3

Determine the equation of the plane that contains the point A = (2, 5, 1) and the line:

Line Equation

Line Equation

Linear Determination

Cartesian Equation of the Plane


4

Find the intersecting point between the plane x + 2y − z − 2 = 0, the line determined by the point (1, −3, 2) and the vector Vector U.

Parametric Equation

Line Equation

Intersecting Point Solution


5

Determine, in intercept form, the equation of the plane that passes through the points A = (2, 0, 0), B = (0, 4, 0) and C = (0, 0, 7).

Intercept Form Equation


6

π is a plane that passes through P = (1, 2, 1) and intersects the positive coordinate semi-axes at points A, B and C. If ABC is an equilateral triangle, determine the equations of π.

Points

Intercept Form Equation

As the triangle is equilateral, the three line segments are equal.

Intercept Form Equation

Equation of the Plane Solution

Plane Solution


7

Find the equation of the plane that passes through the point P = (1, 1, 1) and is parallel to:

Parametric Equation

Linear Determination

Cartesian Equation

Equation of the Plane Solution


8

Determine the equation of the plane that contains the line Line Equation and is parallel to the line Parametric Equation.

The point A = (2, 2, 4) and the vector vector belong to the plane because the line is in the plane.

The vector Vector U is a vector in the plane because it is parallel to the line.

Linear Determination

Equation of the Plane Solution


9

Calculate the equation of the plane that passes through the point (1, 1, 2) and is parallel to the following lines:

Line Equations

System of Equations

System Solution

Parametric Equation

Linear Determination

Equation of the Plane Solution




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