PowerPoint PresentationSlide 2Equations of Lines and PlanesSlide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13PlanesSlide 15Slide 16Slide 17Slide 18Slide 19Copyright © Cengage Learning. All rights reserved. 12Vectors and the Geometry of SpaceCopyright © Cengage Learning. All rights reserved. 12.5Equations of Lines and Planes33Equations of Lines and PlanesLearning objectives:Given information about a line, find a vector equation, parametric equations, and symmetric equations for the line, and equations for a line segmentFind a direction vector of a given lineGiven information about a plane, find a vector equation, scalar equation, and linear equation for the planeFind a normal vector given an equation for a plane, and points on a given plane44Equations of Lines and PlanesA 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.55Equations of Lines and PlanesLet 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 haverepresentations and ).If a is the vector with representation as in Figure 1,then the Triangle Law for vector addition gives r = r0 + a.Figure 166Equations of Lines and PlanesBut, since a and v are parallel vectors, there is a scalar t such that a = t v. Thuswhich is a vector equation of L.Each value of the parameter t gives the position vector rof a point on L. In other words, as t varies, the line is traced out by the tip of the vector r.77Equations of Lines and PlanesIf the vector v that gives the direction of the line L is writtenin 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 becomesx, y, z = x0 + t a, y0 + tb, z0 + t cTwo vectors are equal if and only if corresponding components are equal.88Equations of Lines and PlanesTherefore we have the three scalar equations:where t These equations are called parametric equations of theline L through the point P0(x0, y0, z0) and parallel to thevector v = a, b, c.Each value of the parameter t gives a point (x, y, z) on L.99Equations of Lines and PlanesThe vector equation and parametric equations of a line arenot unique. If we change the point or the parameter orchoose a different parallel vector, then the equations change.1010Equations of Lines and PlanesAnother way of describing a line L is to eliminate theparameter t from Equations 2.If none of a, b, or c is 0, we can solve each of theseequations for t, equate the results, and obtainThese equations are called symmetric equations of L.1111Equations of Lines and PlanesNotice 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. Forinstance, if a = 0, we could write the equations of L asThis means that L lies in the vertical plane x = x0.1212Equations of Lines and PlanesIn 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 + tr1The line segment from r0 to r1 is given by the parameter interval 0 t 1.1313Planes1414PlanesAlthough 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.1515PlanesLet 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 61616PlanesThe normal vector n is orthogonal to every vector in the given plane. In particular, n is orthogonal to r – r0 and so we havewhich can be rewritten asEither Equation 5 or Equation 6 is called a vector equation of the plane.1717PlanesTo obtain a scalar equation for the plane, we writen = a, b, c, r = x, y, z, and r0 = x0, y0, z0. Then the vector equation becomesa, b, c • x – x0, y – y0, z – z0 = 0orEquation 7 is the scalar equation of the plane through P0(x0, y0, z0) with normal vector n = a, b, c.1818PlanesBy collecting terms in Equation 7, we can rewrite the equation of a plane aswhere 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.1919PlanesTwo 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 normal vectors (the angle in Figure 9).(Review Section 12.3 for angles and the dot product.)Figure
View Full Document
Unlocking...