Solving task on-line

The angle between the ribs

Given the coordinates of vertices A₁, A₂, A₃, A₄. By means of vector algebra to find.

The coordinates of the vertices of the pyramid to find:
1) length ribs A₁A₂ and A₁A₃;
2) the angle between the ribs A₁A₂ and A₁A₃;
3) Square face A₁A₂A₃;
4) volume of the pyramid A₁A₂A₃A₄;
5) equation of a line passing through point A₁ and A₂;
6) equation planes A₁A₂A₃ and A₁A₄;
7) angle between the planes A₁A₂A₃ and A₁A₂A₄;
8) equation of the line A₁A₂ and A₁A₃;
9) write vector AB(A₁A₂) and AC (A₁A₃ in the ORT)
10) Equation of the plane passing through the point perpendicular;
11) The length of the height of the pyramid, drawn from the top;
12) The equation of the height of the pyramid over the top;
13) The equation of a line passing through a given point perpendicular to this plane;
14) The distance from point to plane;

Instruction
To solve such problems fill kordinaty vertices, then Next. The resulting solution is stored in the file MS Word. For editing, you can use formul editors Microsoft Equation.
If only three points, the coordinates of A ₄ (D) not fill.

Use designation A, B, C, D

Fill out the coordinates of the vertices

A₁: x y z
A₂: x y z
A₃: x y z
A₄: x y z

Óãîë ìåæäó ðåáðàìè A1A2 A1A3 A1A4 A2A3 A2A4 A3A4 A1A2 A1A3 A1A4 A2A3 A2A4 A3A4

Coordinates of M, dividing A1A2 A1A3 A1A4 A2A3 A2A4 A3A4 on parts: :
(at 1:1 means the division of the segment in half)

The equation of a plane passing through A1 A2 A3 A4 perpendicular vector's A1A2 A1A3 A1A4 A2A3 A2A4 A3A4

The length of the height of the pyramid through the point (the distance from point to plane) A1 A2 A3 A4

Outputs Report:

Square faces A₁A₂A₃
Square faces A₁A₂A₄
Square faces A₁A₃A₄
Square faces A₂A₃A₄
Volume of a pyramid
Dividing the segment in this regard
The equation of line A₁A₂
The equation of line A₁A₃
The equation of line A₁A₄
The equation of line A₂A₃
The equation of line A₂A₄
The equation of line A₃A₄
Equation plane A₁A₂A₃
Equation plane A₁A₂A₄
Equation plane A₁A₃A₄
Equation plane A₂A₃A₄
quation of a plane passing through the point perpendicular
Distance from point to plane (the length of the height of the pyramid)
The equation of straight line through the point perpendicular to the plane (the equation of height through the top of the pyramid)

Little theory

Equation plane, passing through M0(x₀, y₀, z₀) perpendicular's vector N = {A,B,C}, represented as

A(x- x0) + B(y- y0) + C(z- z0) = 0 èëè Ax + By + Cz + D = 0 where D = -Ax₀ - By₀ - Cz₀.
The equation is called the general equation of the plane. Derivation of this equation is the same as finding the general equation of a straight line on plane.

Geometric meaning each letter included in the equation: x,y,z – coordinates of the current point in the plane, x₀, y₀, z₀ – coordinates of the fixed points of the plane, A,B,C – coordinates of any vector perpendicular to the plane, called the normal plane.

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