4 Notes 3

4 Notes 3 - MATH 20C Lecture 5 - Monday, October 4, 2010...

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MATH 20C Lecture 5 - Monday, October 4, 2010 Planes 1. The plane through 3 points, P 1 ,P 2 ,P 3 . A point P = ( x,y,z ) is in the plane if and only if the volume of the parallelipiped with sides --→ P 1 P, ---→ P 1 P 2 , ---→ P 1 P 3 has volume 0 (drew picture). This is the same as saying that det --→ P 1 P ---→ P 1 P 2 ---→ P 1 P 3 = 0 Example Take P 1 = (0 , 1 , 0) ,P 2 = (1 , 1 , 0) ,P 3 = (1 , 0 , 0) . The plane through these 3 points has equation det h x,y - 1 ,z i h 1 , 0 , 0 i h 1 , - 1 , 0 i = 0 which is to say z = 0 . This is the xy -plane. Note : In general the equation of a plane in space has the form ax + by + cz = d 2. The plane through the origin perpendicular to -→ N = h 1 , 5 , 10 i Drew a picture. A point P = ( x,y,z ) is in this plane if and only if --→ OP -→ N, which is to say --→ OP · -→ N = 0 . This means h x,y,z i · h 1 , 5 , 10 i = 0 , which gives x + 5 y + 10 z = 0 . 3. The plane through P 0 = (2 , 1 , - 1) and perpendicular to -→ N = h 1 , 5 , 10 i Drew a picture. A point P = ( x,y,z ) is in this plane if and only if --→ P 0 P -→ N, which is to say --→ P 0 P · -→ N = 0 . This means h x - 2 ,y - 1 ,z + 1 i · h 1 , 5 , 10 i = 0 , which gives x + 5 y + 10 z = - 3 . This plane is parallel to the plane in the previous example. In both cases, the coefficients of x,y,z are the components of the vector -→ N. In the case of x + 5 y + 10 z = - 3 one gets the constant - 3 by plugging in the coordinates of the point P 0 in the left hand side. Definition A vector perpendicular to a plane P is called a normal vector to that plane. Note that this is implies that all normal vectors to a given plane are proportional. So the coefficients of x,y,z are the components of a normal vector to the plane. Conversely, if the equation of the plane is ax + by + cz = d, then h a,b,c, i is a normal vector to it. 1
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Relative positions of lines and planes Two planes: are either parallel (if their normal vectors are proportional) or they intersect in a line. Two lines in space: Here we have 3 possibilities: 1. parallel (so they are in the same plane): same or opposite direction 2. intersect in a point (again they are in the same plane) 3. skew lines (not in the same plane, but no intersection) A line and a plane: To figure it out, take the parametric equation of the line and plug into the equation of the plane. Again, 3 possibilities:
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4 Notes 3 - MATH 20C Lecture 5 - Monday, October 4, 2010...

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