Chapter NineChapter Nine

Vectors and the Geometry Vectors and the Geometry of Spaceof Space

Section 9.1Section 9.1

Three-Dimensional Three-Dimensional

Coordinate SystemsCoordinate Systems GoalsGoals

Become familiar with three-dimensional Become familiar with three-dimensional rectangular coordinates.rectangular coordinates.

Derive the Derive the distance formula in three distance formula in three dimensionsdimensions..

Learn the equation of a Learn the equation of a spheresphere..

Rectangular CoordinatesRectangular Coordinates

To represent points in space, we choose a fixed point O (the origin) and three directed lines through O that are perpendicular to each other.

These are called the coordinate axes and labeled the x-axis, y-axis, and z-axis.

We think of the x- and y-axes as being horizontal and the z-axis as being vertical, and we draw theaxes as shown:

Right-Hand RuleRight-Hand Rule

The direction of the z-axis is determined by the right-hand rule: If you curl the fingers of your right hand

around the z-axis in the direction of a counterclockwise rotation from the positive x-axis to the positive y-axis, then your thumb points in the positive direction of the z-axis:

Coordinate PlanesCoordinate Planes

The coordinate axes determine three coordinate planes. The xy-plane contains the x- and y-axes; The yz-plane contains the y- and z-axes; The xz-plane contains the x- and z-axes.

These coordinate planes divide

space into eight parts, called octants. The first octant, in the foreground, is

determined by the positive axes:

CoordinatesCoordinates

Now if P is any point in space, let… a be the (directed) distance from the yz-plane

to P, b be the distance from the xz-plane to P, and c be the distance from the xy-plane to P.

We represent the point by the ordered triple (a, b, c) of real numbers and we call a, b, and c the coordinates of P.

ProjectionsProjections

The point (The point (aa, , bb, , cc) determines a ) determines a rectangular box.rectangular box.

If we drop a perpendicular from If we drop a perpendicular from PP to the to the xyxy-plane, we get a point -plane, we get a point QQ with with coordinates (coordinates (aa, , bb, 0) called the , 0) called the projectionprojection of of PP on the on the xyxy-plane.-plane.

Similarly for the projections Similarly for the projections RR(0, (0, bb, , cc) & ) & SS((aa, 0, , 0, cc):):

The Space The Space 33

The Cartesian productThe Cartesian product

            = {(= {(xx, , yy, , zz) ) | | xx, , yy, , zz }}

is the set of all ordered triples of is the set of all ordered triples of real numbers and is denoted by real numbers and is denoted by 33..

It is called a It is called a three-dimensional three-dimensional rectangular coordinate systemrectangular coordinate system..

ExampleExample

What surfaces in What surfaces in 33 are represented by the are represented by the following equations?following equations? zz = 3 = 3 yy = 5 = 5

SolutionSolution The equation The equation zz = 3 represents the = 3 represents the set {(set {(xx, , yy, , zz) ) | | zz = 3}, which is the set of all = 3}, which is the set of all points in points in 33 whose whose zz-coordinate is 3.-coordinate is 3.

This is the horizontal plane parallel to the This is the horizontal plane parallel to the xyxy-plane and three units above it.-plane and three units above it.

Solution (cont’d)Solution (cont’d)

The equation The equation yy = 5 represents the = 5 represents the set of all points in set of all points in 33 whose whose yy--coordinate is 5.coordinate is 5.

This is the vertical plane parallel to This is the vertical plane parallel to the the xzxz-plane and five units to the -plane and five units to the right of it.right of it.

ExampleExample

Describe and sketch the surface in Describe and sketch the surface in 33 represented by the equation represented by the equation y = xy = x..

SolutionSolution The equation represents the set The equation represents the set {({(xx, , xx, , zz) ) | | x x , , z z }. This is a vertical }. This is a vertical plane that intersects the plane that intersects the xyxy-plane in the -plane in the line line yy = = xx, , zz = 0. = 0.

The graph shows the portion of this plane The graph shows the portion of this plane that lies in the first octant:that lies in the first octant:

Distance FormulaDistance Formula

We can extend the formula for the We can extend the formula for the distance between two points in a distance between two points in a plane:plane:

For example, the distance from For example, the distance from PP(2, (2, –1, 7) to –1, 7) to QQ(1, –3, 5) is(1, –3, 5) is 2 2 21 2 3 1 5 7 3PQ

Equation of a SphereEquation of a Sphere

We want to find an equation of a sphere We want to find an equation of a sphere with radius with radius rr and center and center CC((hh, , kk, , ll).).

The point The point PP((xx, , yy, , zz) is on the sphere if and ) is on the sphere if and only if only if ||PCPC   || = = r.r. Squaring both sides gives Squaring both sides gives ||PCPC   ||22 = = rr22, or, or

((xx – – hh))22 + + ((yy – – kk))22 + ( + (zz – – ll))22 = = rr22

Equation of a Sphere Equation of a Sphere (cont’d)(cont’d)

For example, we show thatFor example, we show that

xx22 + + yy22 + + zz22 + 4 + 4x – x – 66yy + 2 + 2zz + 6 = 0… + 6 = 0…

……is the equation of a sphere, as well as find its center is the equation of a sphere, as well as find its center and radius:and radius:

SolutionSolution Completing the square gives Completing the square gives

((xx – – hh))22 + + ((yy – – kk))22 + ( + (zz – – ll))22 = = rr22

This is the equation of a sphere with center (– 2, This is the equation of a sphere with center (– 2, 3,– 1) and radius 3,– 1) and radius

8 2 2.

ReviewReview

Three-dimensional rectangular Three-dimensional rectangular coordinate systemcoordinate system

Distance formula in three Distance formula in three dimensionsdimensions

Equation of a sphereEquation of a sphere