Section 13 5 Equations of Lines and Planes

Section 13. 5 Equations of Lines and Planes

VECTOR EQUATION OF A LINE Consider the line L that passes through the point P 0(x 0, y 0, z 0) with direction vector v. Let r 0 be the position vector of point P 0(x 0, y 0, z 0). Then the vector equation of the line L is r = r 0 + tv where r is the position vector for any point (x, y, z) on the line.

PARAMETRIC EQUATIONS OF A LINE Consider the line L that passes through the point P 0(x 0, y 0, z 0) with direction vector Then the parametric equations of the line L are x = x 0 + at y = y 0 + bt z = z 0 + ct

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.

SYMMETRIC EQUATIONS OF A LINE Consider the line L that passes through the point P 0(x 0, y 0, z 0). with direction vector. If none of a, b, or c is 0, then the symmetric equations of the line L are

VECTOR EQUATIONS OF A PLANE Consider the plane passing through the point P 0(x 0, y 0, z 0) with normal vector n. Let r 0 be the position vector of point P 0(x 0, y 0, z 0). Then the vector equation of the plane is n ∙ (r − r 0) = 0 or n ∙ r = n ∙ r 0 where r is the position vector for any point (x, y, z) in the plane.

SCALAR EQUATION OF A PLANE Consider the plane containing the point P 0(x 0, y 0, z 0) with normal vector the scalar equation of the plane is a(x − x 0) + a(y − y 0) + a(z − z 0) = 0 Then

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

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

ANGLE BETWEEN TWO PLANES The angle between two planes with normal vectors n 1 and n 2 is the angle between their normal vectors. To find the angle, use the dot product

DISTANCE BETWEEN A POINT AND A PLANE The distance D between the point P 1(x 1, y 1, z 1) and the plane ax + by + cz + d = 0 is given by