12 Vectors and the Geometry of Space Copyright Cengage Learning All rights reserved 12 5 Equations of Lines and Planes Copyright Cengage Learning All rights reserved Equations of Lines and Planes Learning objectives Given information about a line find a vector equation parametric equations and symmetric equations for the line and equations for a line segment Find a direction vector of a given line Given information about a plane find a vector equation scalar equation and linear equation for the plane Find a normal vector given an equation for a plane and points on a given plane 3 Equations of Lines and Planes A line in the xy plane is determined when a point on the line and the direction of the line its slope or angle of inclination are given The equation of the line can then be written using the point slope form Likewise a line L in three dimensional space is determined when we know a point P0 x0 y0 z0 on L and the direction of L In three dimensions the direction of a line is conveniently described by a vector so we let v be a vector parallel to L 4 Equations of Lines and Planes Let P x y z be an arbitrary point on L and let r0 and r be the position vectors of P0 and P that is they have representations and If a is the vector with representation as in Figure 1 then the Triangle Law for vector addition gives r r0 a Figure 1 5 Equations of Lines and Planes But since a and v are parallel vectors there is a scalar t such that a t v Thus which is a vector equation of L Each value of the parameter t gives the position vector r of a point on L In other words as t varies the line is traced out by the tip of the vector r 6 Equations of Lines and Planes If the vector v that gives the direction of the line L is written in component form as v a b c then we have t v t a t b t c We can also write r x y z and r0 x0 y0 z0 so the vector equation becomes x y z x0 t a y0 tb z0 t c Two vectors are equal if and only if corresponding components are equal 7 Equations of Lines and Planes Therefore we have the three scalar equations where t These equations are called parametric equations of the line L through the point P0 x0 y0 z0 and parallel to the vector v a b c Each value of the parameter t gives a point x y z on L 8 Equations of Lines and Planes The vector equation and parametric equations of a line are not unique If we change the point or the parameter or choose a different parallel vector then the equations change 9 Equations of Lines and Planes Another way of describing a line L is to eliminate the parameter t from Equations 2 If none of a b or c is 0 we can solve each of these equations for t equate the results and obtain These equations are called symmetric equations of L 10 Equations of Lines and Planes Notice that the numbers a b and c that appear in the denominators of Equations 3 are direction numbers of L that is components of a vector parallel to L If one of a b or c is 0 we can still eliminate t For instance if a 0 we could write the equations of L as This means that L lies in the vertical plane x x0 11 Equations of Lines and Planes In general we know from Equation 1 that the vector equation of a line through the tip of the vector r0 in the direction of a vector v is r r0 t v If the line also passes through the tip of r1 then we can take v r1 r0 and so its vector equation is r r0 t r1 r0 1 t r0 tr1 The line segment from r0 to r1 is given by the parameter interval 0 t 1 12 Planes 13 Planes Although a line in space is determined by a point and a direction a plane in space is more difficult to describe A single vector parallel to a plane is not enough to convey the direction of the plane but a vector perpendicular to the plane does completely specify its direction Thus a plane in space is determined by a point P0 x0 y0 z0 in the plane and a vector n that is orthogonal to the plane This orthogonal vector n is called a normal vector 14 Planes Let P x y z be an arbitrary point in the plane and let r0 and r be the position vectors of P0 and P Then the vector r r0 is represented by See See figure 6 Figure 6 15 Planes The normal vector n is orthogonal to every vector in the given plane In particular n is orthogonal to r r0 and so we have which can be rewritten as Either Equation 5 or Equation 6 is called a vector equation of the plane 16 Planes To obtain a scalar equation for the plane we write n a b c r x y z and r0 x0 y0 z0 Then the vector equation becomes a b c x x0 y y0 z z0 0 or Equation 7 is the scalar equation of the plane through P0 x0 y0 z0 with normal vector n a b c 17 Planes By collecting terms in Equation 7 we can rewrite the equation of a plane as where d ax0 by0 cz0 Equation 8 is called a linear equation in x y and z Conversely it can be shown that if a b and c are not all 0 then the linear equation represents a plane with normal vector a b c 18 Planes Two planes are parallel if their normal vectors are parallel For instance the planes x 2y 3z 4 and 2x 4y 6z 3 are parallel because their normal vectors are n1 1 2 3 and n2 2 4 6 and n2 2n1 If two planes are not parallel then they intersect in a straight line and the angle between the two planes is defined as the acute angle between their Figure 9 normal vectors the angle in Figure 9 Review Section 12 3 for angles and the dot product 19
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