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1.3 Lines and Planes

1.3 Lines and Planes. Lines in R 2 and R 3. To determine a line L , we need a point P(x 1 ,y 1 ,z 1 ) on L and a direction vector for the line L. The vector form is. The parametric equations of a line in space is. The parametric equations of a line in the plane is. Example.

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1.3 Lines and Planes

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  1. 1.3 Lines and Planes

  2. Lines in R2 and R3 To determine a line L, we need a point P(x1,y1,z1) on L and a direction vectorfor the line L. The vector form is The parametric equations of a line in space is The parametric equations of a line in the plane is

  3. Example Find vector and parametric equations of the line 1) through two points 2) through the point (2,3,1) and is parallel to

  4. Lines in the Plane To determine a line L, we need a point P(x1,y1) on L and a normal vector that is perpendicular to L. The normal form of the equation of a line in the plane is The general form of the equation is

  5. Equation of a Plane To determine a plane P, we need a point P(x1,y1,z1) on P and a normal vectorthat is orthogonal to P. The normal form of the equation of a plane is The general form of the equation of a plane is • Two planes in space with normal vectors n1 and n2 are either parallel or intersect in a line. • They are paralleliff their normal vectors are. • They are perpendiculariff their normal vectors are.

  6. Equation of a Plane To determine a plane P, we need a point P(x1,y1,z1) on P and twodirection vectors and that are parallel to P. The vector form of the equation of a plane is The parametric form of the equation of a plane is

  7. Examples Find parametric and general forms of the equation of the plane passing 1) through the points 2) through the points (3,2,1), (3,1,-5) and is perpendicular to

  8. Distance • The distance between a plane (with normal vector n) and a point Q (not in the plane) is • where P is any point in the plane. • The distance between a line (with direction vector v) in space and a point Q is where P is any point on the line.

  9. Examples Given two planes with equations 1) Find the distance between the point (1,1,0) and the plane P1. 2) Find the distance between the point (1,-2,4) and the line 3) Show that P1 and P2 are parallel. 4) Find the distance between P1 and P2 .

  10. Distance Formulas • The distance between a plane with equation • and a point Q(x0,y0,z0) (not in the plane) is • The distance between a line in the plane • and a point Q(x0,y0) (not on the line) is

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