1. Equations of Lines and Planes
Section 13.5
Equations of Lines and Planes

2. 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.

3. PARAMETRIC EQUATIONS OF A 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

4. DIRECTION NUMBERS OF A LINE
If is the direction vector for a line, the numbers a, b, and c are called the direction numbers of the line.

5. SYMMETRIC EQUATIONS OF A LINE
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

6. VECTOR EQUATIONS OF 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.

7. SCALAR EQUATION OF A 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

8. GENERAL EQUATION OF A PLANE
The general equation for a plane with normal vector is
ax + by + cz + d = 0.
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

9. PARALLEL PLANES
Two planes are parallel if their normal vectors are parallel.

10. 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

11. 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