Chapter 12 Math 181
An equation of a sphere is
Vectors
Vectors
Cross Product
Lines A line in 3 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 o L. Let P(x, y, z) be an arbitrary point on L and let r0 and r be the position vectors of P0 and P.
But since a and v are parallel vectors, there is a scalar t such that If a is the vector with representation P0P, then the Triangle Law for vector addition gives But since a and v are parallel vectors, there is a scalar t such that Vector Equation of L
As t varies, the line is traced out by the tip of the vector
Parametric Equations of L
where a, b, c are the direction numbers of L Symmetric Equations of L
Planes 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. The orthogonal vector n is called a normal vector. 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 r0 - r is represented by P0P Then the vector r0 - r is represented by P0P. The normal vector n is orthogonal to every vector in the given plane. In particular, n is orthogonal to r0- r.
Scalar Equation of the plane Linear Equation of the plane
Two planes are parallel if their normal vectors are parallel. 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.
Example Find an equation of the plane through the points (3, -1, 2), (8, 2, 4), and (-1, -2, -3).
Example Find an equation of a plane that passes through the point (-1, 2, 1) and contains the line of intersection of the planes
Cylinders A cylinder is a surface that consists of all lines (called rulings) that are parallel to a given line and pass through a given plane curve. When you are dealing with surfaces, it is important to recognize that an equation like represent a cylinder not a circle.
The trace of the cylinder in the xy-plane is the circle
Quadric Surfaces A quadric surface is the graph of a second degree equation in the three variables x, y, z
Ellipsoid All traces are ellipses. If a = b = c, the ellipsoid is a sphere.
Section 13.6 · Table 1a All traces are ellipses. If a = b = c, the ellipsoid is a sphere. Section 13.6 · Table 1a A
Elliptic Paraboloid Horizontal traces are ellipses. Vertical traces are parabolas. Variable raised to the 1st power indicates the axis of the paraboloid.
Elliptic Paraboloid
Hyperbolic Paraboloid Horizontal traces are hyperbolas. Vertical traces are parabolas. The shape resembles a saddle.
Hyperbolic Paraboloid
Cone The horizontal traces are ellipses. The vertical traces in the place x = k and y = k are hyperbolas if k 0 but are pairs of lines if k = 0. The cone opens about the axis of the variable by itself or the variable with the negative sign.
Cone
Hyperboloid of One Sheet Horizontal traces are ellipses. The vertical traces are hyperbolas. The axis of symmetry corresponds to the variable whose coefficient is negative.
Hyperboloid of One Sheet
Hyperboloid of Two Sheets The axis of symmetry corresponds to the variable whose coefficient is positive.
Hyperboloid of Two Sheets
Cylindrical Coordinate System P(r, , z) (r, , 0) x y z
Cylindrical Coordinate System In the cylindrical coordinate system, a point in 3 dimensional space is represented by the ordered triple (r, , z) where r and are the polar coordinates of the projection of P on the xy-plane and z is the directed distance from the xy-plane to P.
Cylindrical Coordinate System
Spherical Coordinate System x y z = c sphere = c half cone = c half plane
Spherical Coordinate System The spherical coordinates (, , ) of a point P in space where = |OP| is the distance from the origin to P, is the same angle as in cylindrical coordinates, and is the angle between the positive z-axis and the line segment OP.
Spherical Coordinate System