1 EquationsofLinesandPlanesin 3-D

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1 1 EquationsofLinesandPlanesin 3-D Recall that given a point P (a, b, c), one can draw a vector from the origin to P. Such a vector is called the position vector of the point P and its coordinates are a, b, c, the same as P. Position vectors are usually denoted r. In this section, we derive the equations of lines and planes in 3-D. We do so by finding the conditions a point P (x, y, z) or its corresponding position vector r = x, y, z must satisfy in order to belong to the object being studied (line or plane).

2 1.1 Lines In 3-D, like in 2-D, a line is uniquely determined when one point on the line and the direction of the line are given. In this section, we assume we are given a point P 0 onthelineandadirection vector v = a, b, c. Our goal is to determine the equation of the line L which goes through P 0 and is parallel to v. Definition 1 a, b, andc are called the direction numbers of the line L. Let P (x, y, z) be an arbitrary point on L. We wish to find the conditions P must satisfy to be on the line L.

3 1.1.1 Vector Equation Figure 1: Line through P 0 parallel to v Consider figure 1. We see that a necessary and sufficient condition for the point P to be on the line L is that P 0 P be parallel to v. This means that there exists a scalar t such that P 0 P = t v Let r 0 be the position vector of P 0 and r be the position vector of P. Then, P 0 P = r r 0

4 Thus, we obtain r r0 = t v That is r = r0 + t v (1) Definition 2 Equation 1 is known as the vector equation of the line L. The scalar t used in the equation is called a parameter. The parameter t can be any real number. As it varies, the point P moves along the line. When t =0, P is the same as P 0. When t>0, P is away from P 0 in the direction of v and when t<0, P is away from P 0 in the direction opposite v.thelargert is (in absolute value), the further away P is from P 0.

5 1.1.2 Parametric Equations If we switch to coordinates, equation 1 becomes x, y, z = x 0,y 0,z 0 + t a, b, c = x 0 + at, y 0 + bt, z 0 + ct Two vectors are equal when their corresponding coordinates are equal. Thus, we obtain x = x 0 + at y = y 0 + bt z = z 0 + ct (2) Definition 3 Equation 2 is known as the parametric equation of the line L Symmetric Equations If we solve for t in equation 2, assuming that a 0, b 0,andc 0we obtain x x 0 a = y y 0 b = z z 0 c (3)

6 Definition 4 Equation 3 is known as the symmetric equations of the line L. Remark 5 Equation 3 is really three equations x x 0 = y y 0 (4) a b y y 0 = z z 0 b c x x 0 = z z 0 a c In the case one of the direction numbers is 0, the symmetric equation simply becomes the equation from 4 which does not involve the direction number being 0. Thevari- able corresponding to the direction number being 0 is simply set to the corresponding coordinate of the given point. For example, if a =0then the symmetric equations are y y 0 b x = x 0 = z z 0 c

7 1.1.4 Examples Example 6 Find the parametric and symmetric equations of the line through P ( 1, 4, 2) in the direction of v = 1, 2, 3 Example 7 Find the parametric and symmetric equations of the line through P 1 (1, 2, 3) and P 2 (2, 4, 1).

8 1.1.5 Intersecting Lines, Parallel Lines Recall that in 2-D two lines were either parallel or intersected. In 3-D it is also possible for two lines not to be parallel and not intersect. Such lines are called skew lines. Example 8 Let L 1 be the line through (1, 6, 2) with direction vector 1, 2, 1 and L 2 be the line through (0, 4, 1) with direction vector 2, 1, 2. Determine if the lines are parallel, if they intersect or if they are skew. If they intersect, find the point at which they intersect. Example 9 Find the angle between the two previous lines. Example 10 Find the points at which L 1 in the example above intersects with the coordinate axes.

9 1.1.6 Summary for Lines 1. Be able to find the equation of a line given a point and a direction or given two points. 2. Be able to tell if two lines are parallel, intersect or are skewed. 3. Be able to find the angle between two lines which intersect. 4. Be able to find the points at which a line intersect with the coordinate planes.

10 1.2 Planes Plane determined by a point and its normal A plane is uniquely determined given a point on the plane and a vector perpendicular to the plane. Such a vector is said to be normal to the plane. To help visualize this, consider figure 2. Given a point P 0 =(x 0,y 0,z 0 ) and a Figure 2: Plane determined by a point and its normal normal n = a, b, c toaplane,apointp =(x, y, z) will be on the plane if P 0 P is perpendicular to n that is n P0 P =0 (5)

11 This is known as the vector equation of a plane. Switchingtocoordinates,weget ax + by + cz + d =0 (6) where d = ax 0 by 0 cz 0. This is known as the scalar equation of a plane. It is also the equation of a plane in implicit form. Remark 11 Note that when we know the scalar equation of a plane, we automatically know its normal; it is given by the coefficients of x, y, andz. Example 12 The scalar equation of a plane through (1, 2, 3) with normal 2, 1, 4 is 2(x 1) + 1 (y 2) + 4 (z 3) = 0 2x + y +4z 16 = 0

12 1.2.2 Plane determined by three points If instead of being given a point and the normal, we are given three non-colinear points P 1, P 2,andP 3,weform the vectors P 1 P 2 and P 1 P 3. The cross product P 1 P 2 P 1 P 3 is a vector perpendicular to both P 1 P 2 and P 1 P 3 and therefore perpendicular to the plane. We can then use one of the point, the vector obtained from the cross product to derive the equation of the plane. Example 13 Find the equation of the plane through the points P 1 (0, 1, 1), P 2 (1, 0, 1) and P 3 (1, 3, 1) Parallel Planes, Intersecting Planes Two planes are parallel if and only if their normals are parallel. If two planes p 1 with normal n 1 and p 2 with normal n 2 are not parallel, then the angle θ between

13 them is defined to be the smallest angle between their normals that is the angle with the non-negative cosine. In other words, θ =cos 1 n1 n 2 n1 n2 (7) Example 14 Findtheanglebetweentheplanesp 1 : x+ y + z 1=0and p 2 : x 2y +3z 1=0. Find their intersection.

14 1.2.4 Summary for Planes In addition, using the material studied so far, you should be able to do the following: 1. Be able to find the equation of a plane given a point ontheplaneandanormaltotheplane. 2. Be able to find the equation of a plane given three points on the plane. 3. Be able to find the equation of a plane through a point and parallel to a given plane. 4. Be able to find the equation of a plane through a point and a line not containing the point. 5. Be able to tell if two planes are parallel, perpendicular.

15 6. Be able to find the angle between two planes. 7. Be able to find the traces of a plane. 8. Be able to find the intersection of two planes. Make sure you can do the above before attempting the problems. 1.3 Problems odd numbers 1-39 and 59 on pages