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Equations of Lines and Planes in 3D Space

Learn about vector and parametric equations of lines, direction numbers, symmetric equations, and vector and scalar equations of planes. Understand parallel planes, angles between planes, and calculating distances between points and planes in 3D space.

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Equations of Lines and Planes in 3D Space

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  1. 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 EQUATIONSOF 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 + aty = y0 + bt z = z0 + ct

  4. 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.

  5. SYMMETRIC EQUATIONSOF 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 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.

  7. SCALAR EQUATIONOF 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 EQUATIONOF 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

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