Chapter 11 Geometry in Space and Vectors

11.1 Cartesian Coordinates in Three-Space

Cartesian Coordinate System

Two Frenchmen, Pierre de Fermat (1601–1665) and René Descartes (1596–1650), introduced what we now call the Cartesian, or rectangular, coordinate system.

• Cartesian/Rectangular coordinate system vs. Polar coordinate system

• Eight Octants in 3-Space

• Right-handed system vs. Left-handed system

The Distance Formula

A parallelepided (i.e., a rectangular box) determined by two points in three-space:

By the Pythagorean Theorem,

Comment: this distance formula in 3-space is nothing but a generalization of distance formula in a plane.

Spheres and Their Equations

Extend from a circle in 2D to a sphere in 3-space:

• A circle is the set of points in a plane that are a constant distance (the radius) from a fixed point (the center).

• A sphere is the set of all points in three-dimensional space that are a constant distance (the radius) from a fixed point (the center).

The standard equation of a sphere:

N.B. A solid sphere is a set of points satisfying \((x-h)^2+(y-k)^2+(z-l)^2\leq r^2\).

The expanded form of the standard equation of a sphere:

N.B. Not every graph of the expanded form (the quadratic equation underneath the box) is a sphere. Conversely, the graph of any equation of this form is either a sphere, a point (a degenerate sphere), or the empty set. In Example 2, if the equation were

N.B. In the 2-space case, the graph of an equation is usually a curve. In 3-space, the graph of an equation was a surface.

The Midpoint formula

If \(P_1(x_1, y_1, z_1)\) and \(P_2(x_2, y_2, z_2)\) are end points of a line segment, then the midpoint \(M(m_1,m_2,m_3)\) has coordinates

That is, simply take the average of corresponding coordinates of the end points.

Planes and Their Equations

The graph of a linear equation in \(x\), \(y\), and \(z\), that is, an equation of the form

is a plane.

Traces

Intersecting the surface of the graph with vertical planes parallel to the coordinate planes, creating a curve on each plane, called a trace.

Coordinate-plane Traces are the lines of intersection of a plane with the coordinate planes.

To draw the traces and sketch the graph of a plane:

1. find the intersection points; that is, find the \(x\)-, \(y\)-, and \(z\)-intercepts,

2. connect these points by line segments to get the traces,

3. shade in the plane.

Curves in 3-space

A review and a generalization of the parametric equation of curves that we learnt in chapter 6.

The arc length of curves in three-space is

Comment: The graph of example 6 is a helix. If we ignore the motion in the z-dimension for a moment, the object is in uniform circular motion. Introducing back the motion in the z-dimension, which is up with constant speed, we see that the object is going around and around as it moves upward, much like a spiral staircase.

Another way to find the length of this curve: peeling a right cylinder.

11.2 Vectors

Representations and Notations for vectors

Scalars vs. Vectors

• Scalars: quantities that can be represented by a single number. e.g., length, mass, volume. Such quantities (and the numbers that measure them) are called scalars.

• Vectors: quantities that require both a magnitude (or length) and a direction for complete specification. e.g., velocity, force, and displacement.

Alternative Terminologies: There is the feather end (the initial point), called the tail, and the pointed end (the terminal point), called the head, or tip.

N.B. Two vectors are considered to be equivalent if they have the same magnitude and direction.

Key Notations Related to the Vectors

• Vectors: \(\mathbf u\) and \(\mathbf v\). Equivalently, \(\vec u\) and \(\vec v\) in handwriting.

• The magnitude (or length): \(\|\vec u\|\) and \(\|\vec v\|\)

  

  which is just the same as the distance formula in 3-space.

• Vector Components in Brackets

• Basis Vectors

Linear Operations on Vectors

• Vector addition is commutative and associative, that is

• Two equivalent ways of adding vectors for finding the sum (or resultant) of vectors

  • the Triangle Law

  • the Parallelogram Law

• Scalar multiplication: \(c\mathbf u\).

  • The negative of \(\mathbf u\): in particular, \(-\mathbf u\), the same length as \(\mathbf u\), but opposite direction.

• Zero Vector: \(-\mathbf u+\mathbf u=\mathbf 0\). \(\mathbf u+\mathbf 0=\mathbf 0+\mathbf u=\mathbf u\)

• Vector Subtraction: \(\mathbf u - \mathbf v=\mathbf u +( - \mathbf v)\)

N.B. The general expression for \(\mathbf m\) can also be written as \[\mathbf u+t(\mathbf v-\mathbf u).\] If we allow \(t\) to range over all scalars, we obtain the set of all vectors with tails at the same point as the tail of \(\mathbf u\) and heads on the line \(l\).

This fact will be important to us later in describing lines using vector language.

An Application Problem in Physics – Force as a Vector

Algebraic Approach to Vectors

Using the components in brackets to represent vectors,

N.B. The vectors \(\mathbf u=<u_1,u_2,u_3>\) and \(\mathbf v=<v_1,v_2,v_3>\) are equal if and only if the corresponding components are equal; that is, \(u_1=v_1\), \(u_2=v_2\) and \(u_3=v_3\).

Basic Three Vector Algebra Operations:

Find the Unit Vector

11.3 Dot Product

Dot Product (or scalar product)

is defined for 3-dimensional vectors as

Properties of the dot product

Angle between the vectors u and v

Orthogonal

Vectors that are perpendicular are said to be orthogonal.

Direction Angles and Cosines

Direction Angle

The smallest nonnegative angles between a nonzero three-dimensional vector \(\mathbf a\) and the basis vectors \(\mathbf i\), \(\mathbf j\), and \(\mathbf k\) are called the direction angles of \(\mathbf a\).

Direction Cosines

It is often more convenient to work with the direction cosines.

N.B.

Projections

Vector Projection of u on v is the vector

Scalar Projection of u on v is the scalar

\(\|\mathbf u\|\cos\theta=\frac{ \|\mathbf u\|\cos\theta\|\mathbf v\|}{\|\mathbf v\|}=\frac{\mathbf u\cdot \mathbf v}{\|\mathbf v\|}\)

N.B. Based on the definition, the scale projection is positive, zero, or negative, depending on whether \(\theta\) is acute, right, or obtuse. However, the length (a.k.a. magnitude) of the vector projection \(\text{pr}_\mathbf v \mathbf u\) is non-negative.

• When \(0\leq \theta \leq \pi/2\), the scalar projection is equal to the magnitude of \(\text{pr}_\mathbf v \mathbf u\), i.e., \(\|\mathbf u\| |\cos\theta|\).

• When \(\pi/2< \theta \leq \pi\), the scalar projection is equal to the opposite of the magnitude of \(\text{pr}_\mathbf v \mathbf u\), that is, \(-\|\mathbf u\| |\cos\theta|\). )

Comment: If we pull on the box with force \(\mathbf u\), the effective force moving the box forward in the direction \(\mathbf v\) is the projection of \(\mathbf u\) onto \(\mathbf v\).

Application in Physics: Work and Force

Fact in Physics

The work done by a constant force \(\mathbf F\) in moving an object along the line from \(P\) to \(Q\) is the magnitude of the force in the direction of the motion, times the distance moved.

If a force \(\mathbf F\) moving an object through a displacement \(\mathbf D = \overrightarrow{PQ}\) has some other direction, the work is performed by the component of \(\mathbf F\) in the direction of \(\mathbf D\). (\(\theta\) is the angle between \(\mathbf F\) and \(\mathbf D\))

Therefore, the formula for the work done is

Planes

The plane through \(P_1\) perpendicular (or normal) to \(\mathbf n\) has vector equation \[\overrightarrow{P_1P}\cdot \mathbf n=0\]

where \(\mathbf n=<A,B,C>\) is a fixed nonzero vector and \(P_1(x_1, y_1, z_1)\) is a fixed point.

Then, for any point \(P(x, y, z)\) in the plane, we can write \(\overrightarrow{P_1P}\) in component form

Now, we obtain the component equation equivalent to \(\overrightarrow{P_1P}\cdot \mathbf n=0\), called the standard form for the equation of a plane,

If we remove the parentheses and simplify, the boxed equation takes the form of the general linear equation

N.B. Every plane has a linear equation. Conversely, the graph of a linear equation in three-space is always a plane.

N.B. Angles between two planes is the angle between their normals.

Using the result in Example 10, we can solve the following problem:

11.4 Cross Product

Cross Product

The cross product \(\mathbf u \times \mathbf v\) of \(\mathbf u =<u_1,u_2,u_3>\) and \(\mathbf v =<v_1,v_2,v_3>\) is defined by

N.B. The cross product of two vectors is a vector.

In order to memorize the formula for the cross product, we need the determinant. Recall a \(3\times 3\) determinant is

Using determinants, the definition formula of the cross product can be written as

The anticommutative law of cross product is

which is true because if we interchange the positions of \(\mathbf u\) and \(\mathbf v\), we interchange the second and third rows of the determinant, and this changes the sign of the determinant’s value.

Geometric Interpretation of \(\mathbf u \times \mathbf v\)

Reflections:

Theorem A tells us the geometric interpretation of the construction of cross product:

• \(\mathbf u \times \mathbf v\) is perpendicular to both \(\mathbf u\) and \(\mathbf v\) and thus to the plane containing them.

• \(\theta\) is the angle between \(\mathbf u\) and \(\mathbf v\), and the fingers of the right hand are curled in the direction of the rotation through \(\theta\) that makes \(\mathbf u\) coincide with \(\mathbf v\).

• Parallel Criterion for Vectors

Applications

• The equation of the plane through three noncollinear points.

• The following examples are two important results:
  • The area of a parallelogram
  • The volume of a parallelepiped

Comments: If the vectors \(\mathbf a\), \(\mathbf b\), and \(\mathbf c\) from the previous example are in the same plane, the parallelepiped has height zero, so the volume should be zero. If \(\mathbf a\) is in the plane determined by \(\mathbf b\) and \(\mathbf c\), then any vector perpendicular to \(\mathbf b\) and \(\mathbf c\) will be perpendicular to \(\mathbf a\) as well. Thus, \(\mathbf a \cdot (\mathbf b \times\mathbf c)=0\).

N.B. The scalar triple product \(\mathbf a \cdot (\mathbf b \times\mathbf c)\) is equal to the determinant of the 3×3 matrix whose rows are the components of \(\mathbf a\), \(\mathbf b\), and \(\mathbf c\). Both the scalar triple product and the determinant measure the signed volume of the parallelepiped formed by these three vectors.

Geometric Interpretation of Scalar Triple Product

The determinant of the 3×3 matrix represents the signed volume of the parallelepiped formed by the vectors \(\mathbf a\), \(\mathbf b\), and \(\mathbf c\). The scalar triple product \(\mathbf a \cdot (\mathbf b \times\mathbf c)\) also represents the same signed volume, because:

• The cross product \(\mathbf b \times\mathbf c\) gives a vector whose magnitude is the area of the parallelogram formed by \(\mathbf b\) and \(\mathbf c\).

• The dot product of this vector with \(\mathbf a\) projects \(\mathbf a\) onto the direction of \(\mathbf b \times\mathbf c\), giving the height of the parallelepiped.

• The product of the area (from \(\mathbf b \times\mathbf c\)) and the height (from the dot product with \(\mathbf a\)) gives the volume.

Properties of Scalar Triple Product

The scalar triple product has the following properties:

• It is cyclic, meaning the order of the vectors can be cyclically permuted without changing the result: \[(\mathbf a \times\mathbf b)\cdot \mathbf c = (\mathbf b \times\mathbf c)\cdot \mathbf a =(\mathbf c \times\mathbf a)\cdot \mathbf b\]

• It is antisymmetric, meaning swapping any two vectors changes the sign: \[(\mathbf a \times\mathbf b)\cdot \mathbf c =-(\mathbf a \times\mathbf c)\cdot \mathbf b\]

Vector Triple Product

Difference \((\mathbf u \times\mathbf v)\times \mathbf w\) and \(\mathbf u \times (\mathbf v\times \mathbf w)\).

N.B. The expression \(\mathbf u \times\mathbf v\times \mathbf w\) is ambiguous because the cross product is not associative. To avoid confusion, we use parentheses to specify the order of operations:

• \((\mathbf u \times\mathbf v)\times \mathbf w\) is distinct from \(\mathbf u \times (\mathbf v\times \mathbf w)\)

• The two expressions have different geometric interpretations and algebraic results:

  • \((\mathbf u \times\mathbf v)\times \mathbf w =(\mathbf u \cdot\mathbf w) \mathbf v-(\mathbf v \cdot\mathbf w) \mathbf u\)

  • \(\mathbf u \times (\mathbf v\times \mathbf w)= (\mathbf u \cdot\mathbf w) \mathbf v-(\mathbf u \cdot\mathbf v) \mathbf w\)

• The geometric interpretations:

  • \((\mathbf u \times\mathbf v)\times \mathbf w\)

    • The cross product \(\mathbf u \times\mathbf v\) is perpendicular to both \(\mathbf u\) and \(\mathbf v\).

    • Taking the cross product of \(\mathbf u \times\mathbf v\) with \(\mathbf w\) brings the result back into the plane spanned by \(\mathbf u\) and \(\mathbf v\).

  • \(\mathbf u \times (\mathbf v\times \mathbf w)\)

    • The cross product \(\mathbf v \times\mathbf w\) is perpendicular to both \(\mathbf v\) and \(\mathbf w\).

    • Taking the cross product of \(\mathbf u\) with \(\mathbf v \times\mathbf w\) brings the result back into the plane spanned by \(\mathbf v\) and \(\mathbf w\).

Further Interpretations:

\[\mathbf u \times (\mathbf v\times \mathbf w)= (\mathbf u \cdot\mathbf w) \mathbf v-(\mathbf u \cdot\mathbf v) \mathbf w\]

Vector triple product identity decomposes the vector \(\mathbf u \times (\mathbf v\times \mathbf w)\) into components that lie in the plane spanned by \(\mathbf v\) and \(\mathbf w\). Specifically:

• \((\mathbf u \cdot\mathbf w) \mathbf v\) is a vector in the direction of \(\mathbf v\), scaled by a dot product.

• \((\mathbf u \cdot\mathbf v) \mathbf w\) is a vector in the direction of \(\mathbf w\), scaled by a dot product.

The result \(\mathbf u \times (\mathbf v\times \mathbf w)\) is a linear combination of \(\mathbf v\) and \(\mathbf w\), and it lies in the plane spanned by \(\mathbf v\) and \(\mathbf w\). (Because taking the cross product of \(\mathbf u\) with \(\mathbf v \times\mathbf w\) brings the result back into the plane spanned by \(\mathbf v\) and \(\mathbf w\).)

Algebraic Properties

The cyclic order of cross product of basis vectors:

N.B. Verify your calculation with \(\mathbf v\cdot (\mathbf u \times \mathbf v)=0\) and \(\mathbf u\cdot (\mathbf u \times \mathbf v)=0\).

11.5 Vector-Valued Functions

A Heuristic Example

Now think of a particle moving in 3-space:

When a particle moves through space during a time interval \(I\), we think of the particle’s coordinates as functions defined on \(I\):

The points \((x, y, z) = (f(t), g(t), h(t)), t\in I\), make up the curve in space that we call the particle’s path. This curve in space can also be represented in vector form: The vector

from the origin to the particle’s position \(P(f(t), g(t), h(t))\) at time \(t\) is the particle’s position vector, which defines \(\mathbf r\) as a vector function of the real variable \(t\) on the interval \(I\).

A Generalization to Vector-valued Functions

• Real-valued (or scalar-valued) functions \(f\) of a real variable \(t\)

• Vector-valued function (or vector function) \(\mathbf F\) of a real variable \(t\)

A Vector-Valued Function

\(\mathbf F\) of a real variable \(t\) associates with each real number \(t\) a vector \(\mathbf{F}(t)\). Thus,

N.B. The use of a boldface letter helps us to distinguish between vector functions and scalar functions.

Calculus for Vector Functions

Limit

\[\lim_{t\to c}\mathbf F(t)=\mathbf L\] intuitively means that the vector \(\mathbf F(t)\) tends toward the vector \(\mathbf L\) as \(t\) tends toward \(c\). Alternatively, it means that the vector \(\mathbf F(t)-\mathbf L\) approaches \(\mathbf 0\) as \(t\to c\).

• Continuity

  As we expect, all the standard limit theorems hold. Also, continuity has its usual meaning; that is,

  • \(\mathbf F\) is continuous if \(\lim_{t\to c}\mathbf F(t)=\mathbf F(c)\)

  • \(\mathbf F\) is continuous at \(c\) if and only if \(f\), \(g\), and \(h\) are all continuous there, from Theorem A.

Derivative \(\mathbf F^\prime (t)\)

The derivative \(\mathbf F^\prime (t)\) is defined just as for real-valued functions by

Written in terms of components,

Here comes the definition formula of derivative in terms of components,

Integral

Integration in terms of components

Curvilinear Motion

Revisit the heuristic example:

We are going to use the vector-valued functions in terms of the parametric equations to study the motion of a point (or a particle) \(P\) in space, thus, the position vector of the point is

As \(t\) varies, the head of \(\mathbf r(t)\) traces the path of the moving point \(P\). This is a curve, and we call the corresponding motion curvilinear motion.

Velocity and Acceleration

The velocity and acceleration of the moving point \(P\) are

\(\mathbf v(t)\) has the direction of the tangent line.The acceleration vector \(\mathbf a(t)\) points to the concave side of the curve (i.e., the side toward which the curve is bending).

Arc Length and the Speed

By the First Fundamental Theorem of Calculus, the derivative of the accumulated arc length is

which is what we think of as speed.

Therefore,

N.B. The speed of an object is a scalar quantity (or a magnitude), whereas its velocity is a vector.

Applications of Curvilinear Motion

Uniform Circular Motion

Suppose that an object moves in the \(xy\)-plane counterclockwise around a circle with center \((0,0)\) and radius \(a\) at a constant angular speed of \(\omega\) radians per second. If its initial position is \((a, 0)\), then its position vector is

Helix

The path traced out by an object whose position vector is given by

is a helix.

Understanding the position vector equation of a helix:

1. If we look at just the \(x\)- and \(y\)-components of motion, we see uniform circular motion.

2. If we look at just the \(z\)-component of motion, we see uniform straight line motion.

3. If we put these two together, we see that the object spirals around and around as it moves higher and higher, that is, a helix.

11.6 Lines and Tangent lines in 3-Space

Parametric Equations of Lines

A line is determined by a fixed point \(P_0\) and a direction vector \(\mathbf v\). It is the set of all points \(P\) such that is parallel to \(\mathbf v\), that is, that satisfy

Then, with the position vectors of \(P_0\) and \(P\) given by \(\mathbf r_0\) and \(\mathbf r\), respectively, we obtain

\[\mathbf{ r}=\mathbf{ r}_0+\overrightarrow{P_0P}\\=\mathbf{ r}_0+t\mathbf v\]

N.B. The direction numbers are not unique; they could be any nonzero constant multiples like \(ka\), \(kb\), and \(kc\).

Symmetric Equations of Lines

Solving each of the parametric equations for \(t\), and equating the results, we obtain the symmetric equations for the line through \((x_0,y_0,z_0)\) with direction numbers \(a,b,c\), which is

The conjunction of the two equations

are the equations of planes.

N.B. The intersection of two planes is a line whose symmetric equation is given in the box above.

Alternatively,

N.B. An alternative solution is based on the fact that the line of intersection of two planes is perpendicular to both of their normals.

The vector \(\mathbf u=<2,-1,-5>\) is normal to the first plane; \(\mathbf v=<4,5,4>\) is normal to the second.

Tangent Line to a Curve

A curve in three-space determine by the position vector

The tangent line to the curve has direction vector

The Plane Perpendicular to a Curve

In order to find plane perpendicular to a smooth curve at a given point \(P\), we need

1. A normal vector for the plane, that is, the direction vector for the tangent line to the curve at \(P\), \(\mathbf r^\prime(t)\);

2. Point \(P\);

3. The standard plane formula in section 11.3, that is equivalent to \(\overrightarrow{P_1P}\cdot \mathbf n=0\). Here \(\mathbf n\) is \(\mathbf r^\prime(t)\).

11.8 Surfaces in Three-Space

The graph of an equation in three variables is normally a surface. So far, we have seen two examples of the graph of an equation in three variables:

• a plane: the graph of \(A x+By+Cz=D\),

• a sphere: the graph of \((x-h)^2+(y-k)^2+(z-l)^2=r^2\)

Cross sections are the intersections of the surface with well-chosen planes.

Comment: The intersections of the surface with the coordinate planes are also called (coordinate) traces.

Use the same trick to find the traces in the xz-plane and the yz-plane, and sketch the graph as following:

Cylinders

A cylinder is a surface that is generated by moving a straight line along a given planar curve while holding the line parallel to a given fixed line. The curve is called a generating curve for the cylinder.

Comment:

In solid geometry, where cylinder means circular cylinder, the generating curves are circles, but now we allow generating curves of any kind, e.g., a hyperbola

in which variable \(z\) is missing.

If \((x_1, y_1, 0)\) satisfies the equation, so does \((x_1, y_1, z)\). As \(z\) runs through all real values, the point traces out a line parallel to the \(z\)-axis. Therefore, the graph of the given equation is a cylinder, a hyperbolic cylinder

Another example of the graph of \(z=\sin y\)

Quadric Surfaces

A quadric surface is the graph in space of a second-degree equation in \(x\), \(y\) and \(z\).

We first focus on quadric surfaces given by the equation

where \(A\), \(B\), \(C\), \(D\) and \(E\) are constants.

The quadric surfaces we now consider have symmetries relative to the \(x\)-, \(y\)-, or \(z\)-axes.

The basic quadric surfaces are ellipsoids, hyperboloids, paraboloids, and elliptical cones.

Comment: Spheres are special cases of ellipsoids.

A Summary Table of Quadric Surfaces

General Quadric Surfaces

In general these surfaces might be translated and rotated relative to the \(x\)-, \(y\)-, or \(z\)-axes. Terms of the type \(Gx\), \(Hy\), or \(Iz\) in the above formula lead to translations, which can be seen by a process of completing the square.

For example,

11.9 Cylindrical and Spherical Coordinates

Alternative ways of specifying the position of a point in 3-space:

• cylindrical coordinate

• spherical coordinate

Cylindrical Coordinate System

Cylindrical coordinates represent a point \(P\) in space by ordered triples \((r,\theta,z)\) in which \(r\geq 0\),

1. \(r\) and \(\theta\) are polar coordinates for the vertical projection of \(P\) on the \(xy\)-plane (\(r\geq 0\), \(0\leq\theta\leq 2\pi\)),

2. \(z\) is the Cartesian (rectangular) vertical coordinate.

Spherical Coordinate System

Spherical coordinates represent a point \(P\) in space by ordered triples \((\rho,\theta,\phi)\) in which

1. \(\rho\) is the distance \(\lvert OP \rvert\) from the origin to \(P\) (\(\rho \geq 0\)),

2. \(\theta\) is the angle from cylindrical coordinates (\(0\leq\theta\leq 2\pi\)),

3. \(\phi\) is the angle line segment \(OP\) makes with the positive \(z\)-axis (\(0\leq \phi\leq \pi\))

Equations Relating to Cartesian Coordinates

Cylindrical and Cartesian coordinates are related by the following equations:

Spherical and Cartesian coordinates are related by the following equations:

With these relationships, we can go back and forth between the two coordinate systems.

Comments: Dividing the equation \(r^2=2r\cos\theta\) by \(r\) appears to lose the solution \(r=0\), i.e., the origin. However, the origin satisfies the equation \(r=2\cos\theta\) with coordinates \((0,\pi/2)\).

We change to Cartesian coordinates by multiplying both sides by \(\rho\).

We substitute for \(x,y,z\)

Application: Spherical Coordinates in Geography

Geographers and navigators use a coordinate system very closely related to spherical coordinates, the longitude-latitude system. By convention,

• longitudes are in degrees east or west of the prime meridian,

• latitudes are in degrees north or south of the equator.

Given \(\rho=3960\) miles, we determine the Cartesian coordinates (as illustrated in Example 4),