Vectors are useful tools for solving two-dimensional problems. Life, however, happens in three dimensions. To expand the use of vectors to more realistic applications, it is necessary to create a framework for describing three-dimensional space. For example, although a two-dimensional map is a useful tool for navigating from one place to another, in some cases the topography of the land is important. Does your planned route go through the mountains? Do you have to cross a river? To appreciate fully the impact of these geographic features, you must use three dimensions. This section presents a natural extension of the two-dimensional Cartesian coordinate plane into three dimensions.
As we have learned, the two-dimensional rectangular coordinate system contains two perpendicular axes: the horizontal x-axis and the vertical y-axis. We can add a third dimension, the z-axis, which is perpendicular to both the x-axis and the y-axis. We call this system the three-dimensional rectangular coordinate system. It represents the three dimensions we encounter in real life.
Definition The three-dimensional rectangular coordinate system consists of three perpendicular axes: the x-axis, the y-axis, and the z-axis. Because each axis is a number line representing all real numbers in the three-dimensional system is often denoted by In (Figure)(a), the positive z-axis is shown above the plane containing the x– and y-axes. The positive x-axis appears to the left and the positive y-axis is to the right. A natural question to ask is: How was arrangement determined? The system displayed follows the right-hand rule. If we take our right hand and align the fingers with the positive x-axis, then curl the fingers so they point in the direction of the positive y-axis, our thumb points in the direction of the positive z-axis. In this text, we always work with coordinate systems set up in accordance with the right-hand rule. Some systems do follow a left-hand rule, but the right-hand rule is considered the standard representation.
In two dimensions, we describe a point in the plane with the coordinates Each coordinate describes how the point aligns with the corresponding axis. In three dimensions, a new coordinate, is appended to indicate alignment with the z-axis: A point in space is identified by all three coordinates ((Figure)). To plot the point go x units along the x-axis, then units in the direction of the y-axis, then units in the direction of the z-axis.go units along the x-axis, then units in the direction of the y-axis, then units in the direction of the z-axis.
Locating Points in Space Sketch the point in three-dimensional space.
To sketch a point, start by sketching three sides of a rectangular prism along the coordinate axes: one unit in the positive direction, units in the negative direction, and units in the positive direction. Complete the prism to plot the point ((Figure)).
Sketch the point in three-dimensional space.
Hint Start by sketching the coordinate axes. Then sketch a rectangular prism to help find the point in space. In two-dimensional space, the coordinate plane is defined by a pair of perpendicular axes. These axes allow us to name any location within the plane. In three dimensions, we define coordinate planes by the coordinate axes, just as in two dimensions. There are three axes now, so there are three intersecting pairs of axes. Each pair of axes forms a coordinate plane: the xy-plane, the xz-plane, and the yz-plane ((Figure)). We define the xy-plane formally as the following set: Similarly, the xz-plane and the yz-plane are defined as and respectively.To visualize this, imagine you’re building a house and are standing in a room with only two of the four walls finished. (Assume the two finished walls are adjacent to each other.) If you stand with your back to the corner where the two finished walls meet, facing out into the room, the floor is the xy-plane, the wall to your right is the xz-plane, and the wall to your left is the yz-plane.
In two dimensions, the coordinate axes partition the plane into four quadrants. Similarly, the coordinate planes divide space between them into eight regions about the origin, called octants. The octants fill in the same way that quadrants fill as shown in (Figure).
Most work in three-dimensional space is a comfortable extension of the corresponding concepts in two dimensions. In this section, we use our knowledge of circles to describe spheres, then we expand our understanding of vectors to three dimensions. To accomplish these goals, we begin by adapting the distance formula to three-dimensional space. If two points lie in the same coordinate plane, then it is straightforward to calculate the distance between them. We that the distance between two points and in the xy-coordinate plane is given by the formulaThe formula for the distance between two points in space is a natural extension of this formula.
The Distance between Two Points in Space The distance between points and is given by the formulaThe proof of this theorem is left as an exercise. (Hint: First find the distance between the points and as shown in (Figure).)and is the length of the diagonal of the rectangular prism having and as opposite corners.
Distance in Space Find the distance between points and
Substitute values directly into the distance formula:
Find the distance between points and
Hint Before moving on to the next section, let’s get a feel for how differs from For example, in lines that are not parallel must always intersect. This is not the case in For example, consider the line shown in (Figure). These two lines are not parallel, nor do they intersect.
You can also have circles that are interconnected but have no points in common, as in (Figure).
We have a lot more flexibility working in three dimensions than we do if we stuck with only two dimensions.
Now that we can represent points in space and find the distance between them, we can learn how to write equations of geometric objects such as lines, planes, and curved surfaces in First, we start with a simple equation. Compare the graphs of the equation in ((Figure)). From these graphs, we can see the same equation can describe a point, a line, or a plane.describes a single point. (b) In the equation describes a line, the y-axis. (c) In the equation describes a plane, the yz-plane.
In space, the equation describes all points This equation defines the yz-plane. Similarly, the xy-plane contains all points of the form The equation defines the xy-plane and the equation describes the xz-plane ((Figure)).describes the xy-plane. (b) All points in the xz-plane satisfy the equation
Understanding the equations of the coordinate planes allows us to write an equation for any plane that is parallel to one of the coordinate planes. When a plane is parallel to the xy-plane, for example, the z-coordinate of each point in the plane has the same constant value. Only the x– and y-coordinates of points in that plane vary from point to point.
Rule: Equations of Planes Parallel to Coordinate Planes
Writing Equations of Planes Parallel to Coordinate Planes
Write an equation of the plane passing through point that is parallel to the xy-plane.
Hint If a plane is parallel to the xy-plane, the z-coordinates of the points in that plane do not vary. As we have seen, in the equation describes the vertical line passing through point This line is parallel to the y-axis. In a natural extension, the equation in describes the plane passing through point which is parallel to the yz-plane. Another natural extension of a familiar equation is found in the equation of a sphere.
Definition A sphere is the set of all points in space equidistant from a fixed point, the center of the sphere ((Figure)), just as the set of all points in a plane that are equidistant from the center represents a circle. In a sphere, as in a circle, the distance from the center to a point on the sphere is called the radius. on the surface of a sphere is units away from the center
The equation of a circle is derived using the distance formula in two dimensions. In the same way, the equation of a sphere is based on the three-dimensional formula for distance.
Rule: Equation of a Sphere The sphere with center and radius can be represented by the equationThis equation is known as the standard equation of a sphere.
Finding an Equation of a Sphere Find the standard equation of the sphere with center and point as shown in (Figure).containing point
Use the distance formula to find the radius of the sphere: The standard equation of the sphere is
Find the standard equation of the sphere with center containing point
Hint First use the distance formula to find the radius of the sphere.
Finding the Equation of a Sphere Let and and suppose line segment forms the diameter of a sphere ((Figure)). Find the equation of the sphere.
Since is a diameter of the sphere, we know the center of the sphere is the midpoint of Then,Furthermore, we know the radius of the sphere is half the length of the diameter. This gives Then, the equation of the sphere is
Find the equation of the sphere with diameter where and
Hint Find the midpoint of the diameter first.
Graphing Other Equations in Three Dimensions Describe the set of points that satisfies and graph the set.
We must have either or so the set of points forms the two planes and ((Figure)).forms the two planes and
Describe the set of points that satisfies and graph the set.
The set of points forms the two planes and
Hint One of the factors must be zero.
Graphing Other Equations in Three Dimensions Describe the set of points in three-dimensional space that satisfies and graph the set.
The x– and y-coordinates form a circle in the xy-plane of radius centered at Since there is no restriction on the z-coordinate, the three-dimensional result is a circular cylinder of radius centered on the line with The cylinder extends indefinitely in the z-direction ((Figure)).This is a cylinder of radius centered on the line with
Describe the set of points in three dimensional space that satisfies and graph the surface.
A cylinder of radius 4 centered on the line with
Hint Think about what happens if you plot this equation in two dimensions in the xz-plane.
Just like two-dimensional vectors, three-dimensional vectors are quantities with both magnitude and direction, and they are represented by directed line segments (arrows). With a three-dimensional vector, we use a three-dimensional arrow. Three-dimensional vectors can also be represented in component form. The notation is a natural extension of the two-dimensional case, representing a vector with the initial point at the origin, and terminal point The zero vector is So, for example, the three dimensional vector is represented by a directed line segment from point to point ((Figure)).is represented by a directed line segment from point to point
Vector addition and scalar multiplication are defined analogously to the two-dimensional case. If and are vectors, and is a scalar, thenIf then is written as and vector subtraction is defined byThe standard unit vectors extend easily into three dimensions as well— and —and we use them in the same way we used the standard unit vectors in two dimensions. Thus, we can represent a vector in in the following ways:
Vector Representations Let be the vector with initial point and terminal point as shown in (Figure). Express in both component form and using standard unit vectors.and terminal point
In component form, In standard unit form,
Let and Express in component form and in standard unit form.
Hint Write in component form first. is the terminal point ofAs described earlier, vectors in three dimensions behave in the same way as vectors in a plane. The geometric interpretation of vector addition, for example, is the same in both two- and three-dimensional space ((Figure)).
We have already seen how some of the algebraic properties of vectors, such as vector addition and scalar multiplication, can be extended to three dimensions. Other properties can be extended in similar fashion. They are summarized here for our reference.
Rule: Properties of Vectors in Space Let and be vectors, and let be a scalar.Scalar multiplication: Vector addition: Vector subtraction: Vector magnitude: Unit vector in the direction of v: ifWe have seen that vector addition in two dimensions satisfies the commutative, associative, and additive inverse properties. These properties of vector operations are valid for three-dimensional vectors as well. Scalar multiplication of vectors satisfies the distributive property, and the zero vector acts as an additive identity. The proofs to verify these properties in three dimensions are straightforward extensions of the proofs in two dimensions.
Vector Operations in Three Dimensions Let and ((Figure)). Find the following vectors.
Let and Find a unit vector in the direction of
Hint Start by writing in component form.
Throwing a Forward Pass A quarterback is standing on the football field preparing to throw a pass. His receiver is standing 20 yd down the field and 15 yd to the quarterback’s left. The quarterback throws the ball at a velocity of 60 mph toward the receiver at an upward angle of (see the following figure). Write the initial velocity vector of the ball, in component form.
The first thing we want to do is find a vector in the same direction as the velocity vector of the ball. We then scale the vector appropriately so that it has the right magnitude. Consider the vector extending from the quarterback’s arm to a point directly above the receiver’s head at an angle of (see the following figure). This vector would have the same direction as but it may not have the right magnitude.
The receiver is 20 yd down the field and 15 yd to the quarterback’s left. Therefore, the straight-line distance from the quarterback to the receiver is We have Then the magnitude of is given byand the vertical distance from the receiver to the terminal point of is Then and has the same direction asRecall, though, that we calculated the magnitude of to be and has magnitude mph. So, we need to multiply vector by an appropriate constant, We want to find a value of so that mph. We haveso we want *** QuickLaTeX cannot compile formula: \begin{array}{}\\ \\ \hfill k\frac{50}{\sqrt{3}}& =\hfill & 60\hfill \\ \hfill k& =\hfill & \frac{60\sqrt{3}}{50}\hfill \\ \hfill k& =\hfill & \frac{6\sqrt{3}}{5}.\hfill \end{array} *** Error message: Missing # inserted in alignment preamble. leading text: $\begin{array}{} Missing $ inserted. leading text: ...n{array}{}\\ \\ \hfill k\frac{50}{\sqrt{3}} Extra }, or forgotten $. leading text: ...n{array}{}\\ \\ \hfill k\frac{50}{\sqrt{3}} Missing } inserted. leading text: ...{array}{}\\ \\ \hfill k\frac{50}{\sqrt{3}}& Extra }, or forgotten $. leading text: ...{array}{}\\ \\ \hfill k\frac{50}{\sqrt{3}}& Missing } inserted. leading text: ...{array}{}\\ \\ \hfill k\frac{50}{\sqrt{3}}& Extra }, or forgotten $. leading text: ...{array}{}\\ \\ \hfill k\frac{50}{\sqrt{3}}& Missing } inserted. leading text: ...{array}{}\\ \\ \hfill k\frac{50}{\sqrt{3}}& Extra }, or forgotten $. leading text: ...{array}{}\\ \\ \hfill k\frac{50}{\sqrt{3}}& Missing } inserted. Then Let’s double-check that We haveSo, we have found the correct components for
Assume the quarterback and the receiver are in the same place as in the previous example. This time, however, the quarterback throws the ball at velocity of mph and an angle of Write the initial velocity vector of the ball, in component form.
Hint Follow the process used in the previous example.
Consider a rectangular box with one of the vertices at the origin, as shown in the following figure. If point is the opposite vertex to the origin, then find
a. b.
Find the coordinates of point and determine its distance to the origin.
For the following exercises, describe and graph the set of points that satisfies the given equation.
A union of two planes: (a plane parallel to the xz-plane) and (a plane parallel to the xy-plane)
A cylinder of radius centered on the line
Write the equation of the plane passing through point that is parallel to the xy-plane.
Write the equation of the plane passing through point that is parallel to the xz-plane.
Find an equation of the plane passing through points and
Find an equation of the plane passing through points andFor the following exercises, find the equation of the sphere in standard form that satisfies the given conditions.
Center and radius
Center and radius
Diameter where and
Diameter where andFor the following exercises, find the center and radius of the sphere with an equation in general form that is given.
Center and radius
For the following exercises, express vector with the initial point at and the terminal point at
and
a. b.
and
and where is the midpoint of the line segment
a. b.
and where is the midpoint of the line segment
Find terminal point of vector with the initial point at
Find initial point of vector with the terminal point atFor the following exercises, use the given vectors and to find and express the vectors and in component form.
For the following exercises, vectors u and v are given. Find the magnitudes of vectors and
where is a real number.
where is a real number. For the following exercises, find the unit vector in the direction of the given vector and express it using standard unit vectors.
where and
where
where and
where and
Determine whether and are equivalent vectors, where and
Determine whether the vectors and are equivalent, where andFor the following exercises, find vector with a magnitude that is given and satisfies the given conditions.
and have the same direction
and have the same direction
and have opposite directions for any where is a real number
and have opposite directions for any where is a real number
Determine a vector of magnitude in the direction of vector where and
Find a vector of magnitude that points in the opposite direction than vector where and Express the answer in component form.
Consider the points and where and are negative real numbers. Find and such that
Consider points and where and are positive real numbers. Find and such that
Let be a point situated at an equal distance from points and Show that point lies on the plane of equation
Let be a point situated at an equal distance from the origin and point Show that the coordinates of point satisfy the equation
The points and are collinear (in this order) if the relation is satisfied. Show that and are collinear points.
Show that points and are not collinear.
[T] A force of acts on a particle in the direction of the vector where
a. b.
[T] A force of acts on a box in the direction of the vector where
If is a force that moves an object from point to another point then the displacement vector is defined as A metal container is lifted m vertically by a constant force Express the displacement vector by using standard unit vectors.
A box is pulled yd horizontally in the x-direction by a constant force Find the displacement vector in component form.
The sum of the forces acting on an object is called the resultant or net force. An object is said to be in static equilibrium if the resultant force of the forces that act on it is zero. Let and be three forces acting on a box. Find the force acting on the box such that the box is in static equilibrium. Express the answer in component form.
[T] Let be forces acting on a particle, with
The force of gravity acting on an object is given by where m is the mass of the object (expressed in kilograms) and is acceleration resulting from gravity, with A 2-kg disco ball hangs by a chain from the ceiling of a room.
a. N; b. N
A 5-kg pendant chandelier is designed such that the alabaster bowl is held by four chains of equal length, as shown in the following figure.
[T] A 30-kg block of cement is suspended by three cables of equal length that are anchored at points and The load is located at as shown in the following figure. Let and be the forces of tension resulting from the load in cables and respectively.
a. N; b. and (each component is expressed in newtons)
Two soccer players are practicing for an upcoming game. One of them runs 10 m from point A to point B. She then turns left at and runs 10 m until she reaches point C. Then she kicks the ball with a speed of 10 m/sec at an upward angle of to her teammate, who is located at point A. Write the velocity of the ball in component form.
Let be the position vector of a particle at the time where and are smooth functions on The instantaneous velocity of the particle at time is defined by vector with components that are the derivatives with respect to of the functions x, y, and z, respectively. The magnitude of the instantaneous velocity vector is called the speed of the particle at time t. Vector with components that are the second derivatives with respect to of the functions and respectively, gives the acceleration of the particle at time Consider the position vector of a particle at time where the components of are expressed in centimeters and time is expressed in seconds.
a. (each component is expressed in centimeters per second); (expressed in centimeters per second); (each component expressed in centimeters per second squared);b.
[T] Let be the position vector of a particle at time (in seconds), where (here the components of are expressed in centimeters).
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