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Equations of a plane: general, normal, intercept and three-point forms. Equations of a line: parametric, symmetric and two-point form. Traces, intercepts, pencils.

Trace. If a line, plane or any surface in space intersects a coordinate plane, the point, line, or curve of intersection is called the trace of the line, plane or surface on that coordinate plane.

Intercept. The intercept of a straight line, curve, or surface on a coordinate axis is the distance from the origin to the point where the line, curve, or surface cuts the axis. The intercept on the x-axis is the x -intercept, the intercept on the y-axis is the y-intercept, and the intercept on the z-axis is the z-intercept.

Example. The intercepts of the line 2x + 3y = 6 on the x-axis and y-axis, respectively, are 3 and 2.

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*PLANES*

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Equation of a plane. In an x-y-z Cartesian coordinate system the general form of the equation of a plane is

ax + by + cz + d = 0 .

It is an equation of the first degree in three variables.

The locus of any equation of the first degree in three variables is a plane in three-dimensional Cartesian space. Conversely, any plane in space can be represented by some equation of the first degree in three variables. If one of the variables is missing (that is, if one of the coefficients a, b, c is zero), the plane is parallel to the axis of the missing variable. If two of the variables are missing, the plane is parallel to the plane of the missing variables.

Example. The equation z = k represents a plane parallel to the xy plane and k units from it.

Special forms of the equation of a plane:

1) Intercept form of the equation of a plane. The intercept form of the equation of a plane is

where a, b, and c are the x, y, and z intercepts, respectively (all intercepts assumed to be non-zero).

2) Three point form of the equation of a plane. The equation of a plane passing through three
non-collinear points (x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2}), (x_{3}, y_{3}, z_{3}) is

Evaluation by minors using the top row shows that it is of the first degree and thus represents a plane. It can also be seen that it is satisfied by the coordinates of each of the given points because if the coordinates of any one of the points is substituted in place of x, y and z, the determinant will contain two rows that are just alike, and so will vanish.

3) Normal or perpendicular form of the equation of a plane. A plane is determined if the length and the direction of the perpendicular line from the origin to the plane are known. The normal form of the equation of a plane is

1) λx + μy + νz - p = 0

where λ, μ, and ν represent the direction cosines of a normal to the plane directed away from the origin and p is the perpendicular distance from the origin to the plane.

This equation is best understood in its vector version. Let OQ be a line extending from the origin and perpendicular to plane C, intersecting plane C at Q, and of length p. See Fig. 1. Let n be a unit vector emanating from the origin and extending along line OQ. Let r be a position vector of a variable point P on plane C. Then

r • n = p

In words, this states the condition that the dot product of r times n is p which means that the projection of the vector r on the unit normal n is p. Here n = (λ, μ, ν) and r = (x, y, z). Thus r • n = p is equivalent to λx + μy + νz = p and we have an interpretation for 1) above.

4) Normal and one point form of the equation of a plane. If a plane passes through point (x_{0},
y_{0}, z_{0}) and has a normal with direction numbers a, b, c its equation is

2) a(x - x_{0}) + b(y - y_{0}) + c(z - z_{0}) = 0

This equation is also best understood in its vector version. Let P_{0} represent the point (x_{0}, y_{0}, z_{0}),
P represent the point (x, y, z), and n represent the vector (a, b, c). Now define vector S as

S = P - P_{0} = (x-x_{0}, y-y_{0}, z-z_{0})

Then

S • n = 0 .

In words, this states that the dot product of S and n is zero, which is the condition for
perpendicularity of two vectors. Two vectors are perpendicular if and only if their dot product is
zero. Since S • n = 0 is simply the vector form of a(x - x_{0}) + b(y - y_{0}) + c(z - z_{0}) = 0 it gives
an interpretation to 2).

Normal form of a plane in terms of the direction numbers of the normal. Let a plane lie at a perpendicular distance p from the origin. Let l, m, n be any set of direction numbers of the normal to the plane. Then

and the normal form of the equation of a plane becomes

where the sign of the radical can be determined only if we know on which side of the origin the plane passes.

Converting an equation in general form to normal form. Any equation of the general form

ax + by + cz + d = 0

can be expressed in the normal form by dividing it by

where the sign of the radical must be taken opposite to the sign of d so that the value of p will be positive. Thus ax + by + cz + d = 0 becomes

which is the equation of a plane whose normal has direction numbers a, b, c and whose distance from the origin is

Direction numbers for the normal to a plane. A set of direction numbers for the normal to the plane

ax + by + cz + d = 0

are the coefficients a, b, c.

Theorem. The coefficients a, b, c in the equation

ax + by + cz + d = 0

are direction numbers of the normal to the plane ax + by + cz + d = 0 . Conversely, any set of direction numbers of the normal to a plane can be used as the coefficients of x, y, and z in writing the equation of the plane.

Perpendicular distance of a point from a plane. Let D denotes the distance
from the plane ax + by + cz + d = 0, where d
0, to the point P_{1}(x_{1}, y_{1}, z_{1}). If this distance is
counted positive when P_{1} lies on the opposite side of the plane from the origin and negative when
it lies on the same side of the plane as the origin, then

where the sign of the radical is opposite to that of d.

Angle between two planes. The dihedral angle between two planes is measured by the
angle between normals to the planes. The acute angle between planes a_{1}x + b_{1}y + c_{1}z + d_{1} = 0
and a_{2}x + b_{2}y + c_{2}z + d_{2} = 0 is given by

The two planes are perpendicular if and only if

a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0 .

The two planes are parallel if and only if

Direction of line of intersection of two planes. A set of direction numbers for
the line of intersection of the planes a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 and a_{2}x + b_{2}y + c_{2}z + d_{2} = 0 is

Equation of plane through point P_{1}(x_{1}, y_{1}, z_{1}) and parallel to
directions (a_{1}, b_{1}, c_{1}) and (a_{2}, b_{2}, c_{2}). A plane through point P_{1}(x_{1}, y_{1}, z_{1}) and
parallel to directions (a_{1}, b_{1}, c_{1}) and (a_{2}, b_{2}, c_{2}) is given by

Equation of plane through point P_{1}(x_{1}, y_{1}, z_{1}) and P_{2}(x_{2}, y_{2}, z_{2}) and
parallel to direction (a, b, c). A plane through point P_{1}(x_{1}, y_{1}, z_{1}) and P_{2}(x_{2}, y_{2}, z_{2})
and parallel to direction (a, b, c) is given by

Def. Pencil of planes. All the planes passing through a given line. The line is the axis of the pencil.

Theorem 1. Let II_{1} = 0 and II_{2} = 0 be the equations of two intersecting planes where II_{1} and
II_{2} are linear expressions in x, y, and z. Then the family of planes

k_{1}II_{1} + k_{2}II_{2} = 0 ,

where k_{1} and k_{2} are parameters (arbitrary constants), is the pencil of planes passing through the
intersection of the planes II_{1} = 0 and II_{2} = 0 .

Theorem 2. Four points (x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2}), (x_{3}, y_{3}, z_{3}), and (x_{4}, y_{4}, z_{4}) lie in the same plane
if

Theorem 3. Four planes

a_{1}x + b_{1}y + c_{1}z + d_{1} = 0

a_{2}x + b_{2}y + c_{2}z + d_{2} = 0

a_{3}x + b_{3}y + c_{3}z + d_{3} = 0

a_{4}x + b_{4}y + c_{4}z + d_{4} = 0

pass through the same point if

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*LINES*

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Equations of a line. A line is determined in three principal ways:

1) By the intersection of two planes

2) By two points on the line

3) By a point and a set of direction numbers

Two-point form of equations of a line. If a line passes through the points P_{1}(x_{1}, y_{1}, z_{1}) and
P_{2}(x_{2}, y_{2}, z_{2}) the equations for the line can be written as

Symmetric form of the equations for a line. If a line passes through the point P_{1}(x_{1}, y_{1}, z_{1}) with
direction numbers l, m, n the equations for the line can be written as

Parametric form of the equations of a line. If a line passes through the point P_{1}(x_{1}, y_{1}, z_{1}) with
direction numbers l, m, n the equations for the line can be written in parametric form as

x = x_{1} + lt

y = y_{1} + mt

z = z_{1} + nt .

As the parameter t varies through all real numbers these equations give the coordinates of all the
points and only the points that lie on the line. Positive values of t correspond to points on one
side of P_{1}(x_{1}, y_{1}, z_{1}), negative values of t to points on the other side, and the value zero to the
point P_{1}.

If a line passes through the point P_{1}(x_{1}, y_{1}, z_{1}) with direction cosines λ, μ, and ν the equations for
the line can be written as

x = x_{1} + λt

y = y_{1} + μt

z = z_{1} + νt .

In this case the parameter t has special significance. It is the actual directed distance from P_{1} to
P(x,y,z) along the line. To see this consider the vector form of the system where U corresponds
to the unit vector U = ( λ, μ, ν ), X = (x, y, z) and X_{1} = (x_{1}, y_{1}, z_{1}). The vector form then is

X = X_{1} + Ut .

If a line passes through two points P_{1}(x_{1}, y_{1}, z_{1}) and P_{2}(x_{2}, y_{2}, z_{2}) the equations of the line can be
written

x = x_{1} + (x_{2} - x_{1})t

y = y_{1} + (y_{2} - y_{1})t

z = z_{1} + (z_{2} - z_{1})t

The point P_{1} corresponds to t = 0 and P_{2} to t = 1.

Distance between a point and a line. Let P_{0}(x_{0}, y_{0}, z_{0}) be any point and let L be a
line passing through a point P_{1}(x_{1}, y_{1}, z_{1}) in direction (a, b, c). Let d be the perpendicular
distance between P_{0} and line L. Then the distance d is given by

Distance between two skew lines. Let L_{1} be a line passing through point P_{1}(x_{1}, y_{1},
z_{1}) in direction (a_{1}, b_{1}, c_{1}) and L_{2} be a line passing through point P_{2}(x_{2}, y_{2}, z_{2}) in direction (a_{2}, b_{2},
c_{2}). Then the perpendicular distance d between the two lines is given by

References.

Samuel Selby. CRC Standard Mathematical Tables

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