Section 13.5

Equations of Lines and Planes

VECTOR EQUATION OF A LINE

Consider the line L that passes through the point P0(x0, y0, z0) with direction vector v. Let r0 be the position vector of point P0(x0, y0, z0). Then the vector equation of the line L is

r = r0 + tv

where r is the position vector for any point (x, y, z) on the line.

Consider the line L that passes through the point P0(x0, y0, z0) with direction vector Then the parametric equations of the line L are

x = x0 + at y = y0 + bt z = z0 + ct

PARAMETRIC EQUATIONSOF A LINE

.,, cbav

DIRECTION NUMBERSOF A LINE

If is the direction vector for a line, the numbers a, b, and c are called the direction numbers of the line.

cba ,,v

Consider the line L that passes through the point P0(x0, y0, z0). with direction vector . If none of a, b, or c is 0, then the symmetric equations of the line L are

SYMMETRIC EQUATIONSOF A LINE

cba ,,v

c

zz

b

yy

a

xx 000

VECTOR EQUATIONSOF A PLANE

Consider the plane passing through the point the point P0(x0, y0, z0) with normal vector n. Let r0 be the position vector of point P0(x0, y0, z0). Then the vector equation of the plane is

n ∙ (r − r0) = 0

or

n ∙ r = n ∙ r0

where r is the position vector for any point (x, y, z) in the plane.

Consider the plane containing the point P0(x0, y0, z0) with normal vector Then the scalar equation of the plane is

a(x − x0) + a(y − y0) + a(z − z0) = 0

SCALAR EQUATIONOF A PLANE

.,, cban

The general equation for a plane with normal vector is

ax + by + cz + d = 0.

GENERAL EQUATIONOF A PLANE

.,, cban

cba ,,n

This equation is called a linear equation in x, y, and z. If a, b, and c are not all zero, then the linear equation represents a plane with normal vector

PARALLEL PLANES

Two planes are parallel if their normal vectors are parallel.

ANGLE BETWEEN TWO PLANES

The angle between two planes with normal vectors n1 and n2 is the angle between their normal vectors. To find the angle, use the dot product

|n||n|

nn

21

21 cos

DISTANCE BETWEEN A POINT AND A PLANE

The distance D between the point P1(x1, y1, z1) and the plane ax + by + cz + d = 0 is given by

222

111 ||

cba

dczbyaxD