Merced College; Don Power

MULTIVARIATE CALCULUS - CH 12, LECTURE

THREE-DIMENSIONAL SPACE; VECTORS

12.1 Rectangular Coordinates in 3-Space; Spheres; Cylindrical Surfaces

3D coord system: depicts ordered triples;

Why? consider a surface; height of the surface can depend on position on the plane

(z can be a function of both x and y)

Demo: handkerchief over just desk, then over book+wallet+keys.

So 3D space is R³, 2D plane is R², and the real number line (1D) is R.

How? Coord axes: typically x,y for horizontal plane, z for vertical axis

Paper/pencil representations:

Octants: "first octant." is the only one that is labeled. (everything is pos.)

How to graph ordered triples without ambiguity: Ex: (2, 3, 4) vs (0,2,2) on 2nd system above.

Projection of this point onto xy-plane (or xz-plane, or yz-plane)

RH vs LH coord syst

Demo: Hold pencil (x+), pen (y+), red pen (z+), guess which orientation.

Paper/pencil representations (write labels, then guess)

We'll usually see RH systems.

Axes:

What set of ordered triples represent the x-axis? Points of form (x,0,0); Other axes?

Equation - will have to wait [requires parametric equations]

Coord planes

What set of ordered triples represents the xy-plane? Points of form (x,y,0)

For xy-plane, z=0 at every point, so z=0 is the eqn of the xy-plane. Other coord planes?

Equations of planes parallel to coordinate planes:

Ex: If it’s parallel to the xz-plane, it crosses the y-axis at some point y₀. So equ is y=y₀.

Cylinders

The familiar cylinder is a right circular cylinder. More generally, a cylinder is formed by a generating curve in a plane, projected along parallel lines not in the same plane. If the plane is a coord plane, then the curve is an eqn in two of the variables (x, y, z), and that same eqn is the eqn of the cylinder.

How to graph a plane like y=-3, z=-x

Draw parallel "traces" to form a parallelogram representing a rectangle;

The brain interprets the result as a rectangle in the appropriate orientation.

Extensions of 2D formulas:

Distance formula in 3D (and notation |P₁P₂|): d = sqrt((x₂-x₁)²+(y₂-y₁)²+(z₂-z₁)²)

Sphere: (x-x₀)²+(y-y₀)²+(z-z₀)²=r²

Midpoint formula: Each coordinate is the mean of the corresponding coords of the two points.

Ex: Find equation of sphere for which the endpoints of a diameter are (1,-4,-3) and (-5,-6,7)

Ex: Find center and radius of sphere tangent to the xz-plane if farthest point from the plane is (2,-3,-4)

Ex: Describe the region in R³ rep by the inequality x²+ z² -6z < -5. Note y is arbitrary.

Is the boundary of the region included?

Sample test questions:

1. Complete the square to find the center and radius of the sphere x²+y²+z²+10x+4y+2z-19=0.

2. Write an equation of the sphere whose center is (3,-2,-5) that passes through (2,1,-1)

12.2 Vectors

Def: A vector is a number that has both magnitude and direction.

(Contrast real numbers, or "scalars," -- magnitude only, except for the sign)

Geometrically, represent with

Arrow from initial point P to terminal point Q, (displacement vector - change of position)

An arrow from the origin to a point P ("standard position") , or

An arrow in an arbitrary location (in general, a vector can be shown in any location)

Ex: Force vectors indicate magnitude and direction of a applied forces

Ex: Velocity vectors indicate speed and direction of movement along various paths.

Notation:

See the above, if the vector links two labeled points.

Otherwise, we use lower case bold letter (in print), or overbar (handwritten), v, or .

Two vectors are equal if they have the same magnitude and direction (location is irrelevant)

Def: Sum v + w is a vector u such that if v and w are head-to-tail, u is from the tail of v to the head of w

i.e. sum vector is along the diagonal of a parallelogram determined by the two vectors.

Ex: [subtraction] How can you represent v-w?

Def: scalar multiple kv

Coordinate form in two or 3 dimensions <v₁,v₂>, or <v₁,v₂,v₃>

Ex: coord form of the 0 vector

Equivalent vectors: corresp coords are equal

Ex Solve for r: <1,2,3> = <q,r,s>

Sums and scalar multiples in coord form

Calculation of a displacement vector in component form in 2- or 3-space:

Principle: terminal coordinates - initial coordinates

Ex: Find the vectors and if P is the vector <1,-3,5> and Q is <7,4,-1>

Thm 12.2.6: Field properties applied to vectors: commutative, 2 assoc, 2ident, inverse+, 2 distrib.

How to prove? Component-wise (text does for assoc+)

Def/Notation: Magnitude or norm: ||v||

Calculation by Pythag thm / distance formula

Thm: ||kv|| = k ||v||

Standard unit vectors i, j, k = <1,0,0> etc

Write <1,2,3> as i + 2j + 3k

Normalizing a vector --i.e. finding a unit vector in the direction of a given vector:

Divide the vector by its own norm 1/||v|| times v, or v / ||v||

Ex: Normalize ...

Ex: Find a vector of length 6 in the direction of ...

Maple:

Definition of vectors,

Linear combination,

Norm,

Unit Vector:

or equivalently,

Angle φ with x axis:

unit vector in direction of v is <cos φ, sin φ>

so v is ||v||<cos φ,sin φ> = ||v||cos φ i + ||v||sin φ j

Find a vector in 2-space of length ... that makes an angle of ... with the x-axis

Resultant of forces F1 and F2 at an angle φ (with F1): Find ||F1+F2|| and the resultant angle α (with F1)

setup

label parallelogram with F1 as base, F1+F2 as diagonal, α between them, p-φ opposite F1+F2.

norm by law of cosines:

||F1+F2|| = ||F1||² + ||F2||² +2 ||F1|| ||F2|| cos φ since cos(p-φ) = -cos φ

angle by law of sines:

sin α = ||F2|| sin φ / ||F1+F2||

Sample questions:

Let u = 4i-2j+7k and v = -2i+j+k. Find

--. x, where 3u+x = v

--.

--. A vector of length 5 in the direction of v (Hint: 5 times a unit vector in the direction of v).

--. Scalars a and b such that au+bv = -2i+j+28k.

12.3 DOT PRODUCT

Calculation of u dot v in terms of u₁v₁ + u₂v₂ + ...

Properties

Commutative

Distributive

k(u dot v): the scalar can multiply either vector (it doesn't distribute to both)

0 vector dot v = 0 scalar

unexpected: v dot v = (norm of v)²

Proof (component-wise, for R3)

Let v = <v₁,v₂,v₃>, calculate LHS and RHS, equate

Angle between vectors:

cosq = dot product / product of norms

Proof (in text) is by law of cosines

Ex: Ex 2a in text:

Note that a negative cosine gives us an obtuse angle (cosq is neg if q is in second quadrant)

INTRODUCE LAB1

Ex: Check base angle in tetrahedron, between <2,0,0> and (<1,sqrt(3),0>

Note: a good cross-check on calculation of upper vertex coord is to verify angle between edges

Cosines of direction angles (between vector and coord axes):

Take dot product with i, j, k

Result: cos α = v₁ / norm(v),

cos β = v₂ / norm(v),

cos γ = v₃ / norm(v)

Ex for a vector in R3

Decomposing a vector into a sum of orthogonal components:

Let e₁, e₂ be orthogonal (perpendicular) unit vectors...

Then the component of v in the direction of each unit vector has magnitude v dot e₁ or v dot e₂

and direction e₁ or e₂ respectively

Note from trig: if q is the angle between v and e1, then

v dot e1 = norm(v) cosq

v dot e2 = norm(v) sinq

Vector projection onto

unit vector e: magnitude is always v dot e; direction is e

arbitrary vector b: magnitude is v dot b/norm(b); direction is b/norm(b)

See Ex 6 and go over notation proj_vb and v- proj_vb

Work = Fd = F dot v where v is a vector giving the distance and direction of the motion of the object

See Ex 7

12.4 CROSS PRODUCT

Calculation of

2X2 determinant

3X3 determinant by expansion across the first row

Key properties (easily demonstrated for 2X2)

If two rows are equal, det is 0

Row swap changes the sign

Def of cross product in terms of a determinant

Ex: calculate a sample cross product

Thm 12.4.3 Properties of cross-product

a. Anticommutative

b. L and R Distrib

c. Scalar can apply to either factor

d. Identity is 0 vector

e. Cross prod of a vector with itself is 0

Cross prods of unit vectors

Illustrate with circuit i..j..k..and back to i

Thm 12.4.4 plus

Cross prod of two vectors is orthog to both vectors, with direction given by right-hand rule

See symbolic form as given in the text

Thm 12.4.5

Cross-prod formula for sinq Proof in text

Area of parallelogram Follows from triangle trig

Cross prod is 0 for parallel vectors

Scalar triple product

Volume of parallelepiped

Test for coplanar vectors

12.5 PARAMETRIC EQUATIONS OF LINES

Eqn of line through P₀ (x₀,y₀,z₀) parallel to vector v = <a,b,c>, with P (x,y,z) an arbitrary point on line.

1. Vector forms:

Concept vector P₀P is parallel to v

P₀P = tv for parameter t; as t ranges over real nrs, you get every point on the line

Expand: <x-x₀,y-y₀,z-z₀> = t <a,b,c>

Solve: <x,y,z> = <x₀,y₀,z₀> + t <a,b,c> Easiest form for me to remember

Interpret: Get to arbitrary point (x,y,z) by starting at fixed point and adding scalar mult of v.

2. Parametric eqns: Get eqn for each component in vector form:

x = x₀ + at

y = y₀ + bt

z = z₀ + ct

3. Symmetric eqns: Solve all for t, equate: (x-x₀) / a = (y-y₀) / b = (z-z₀) / c

Example: for some arbitrary P₀ = ( ___ ) , v = < ___ >

Parallel lines: vector portions <a,b,c> are scalar multiples of one another

Intersecting lines: Simultaneous eqns, but you must make sure to use different letters (or subscripts) for the parameters for the two lines.

Ex (adaptation of #25): L1: x = 2 + t, y = 2 + 3t, z = 3 + t

L2: x = 4t + 1, y = 6t + 5, z = 2t + 4. Replace t in L2's eqns by u

Solve simult eqns for x and y, then use the values of t and u to calculate z

If both z's are the same, lines intersect.

If the z's differ, lines are skew.

Line segment joining two points:

Where does v come from?

Specify values for parameter t (0 and 1 if built in the standard manner) to designate endpoints.

Vector form of eqn in text: r = r₀ + tv

Relate to form we started with

Use of r for "vector valued function"

Sample test questions:

Let u = 2i-j+k, v = 3i+3j-k, w = i+4j-2k. Find

The cosine of the angle between u and v. [This checks dot products]

A unit vector in the direction of w.

The vector component of v in the direction of w (i.e. the scalar projection of v onto w)

proj_wv.

Determine whether u and w are orthogonal, parallel, or neither.

A vector orthogonal to both u and v. [This checks cross products]

The area of the triangle determined by u and v.

The volume of the oblique box (parallelepiped) determined by u, v, and w.

A unit vector in the xy-plane that is perpendicular to w.

[What vector would you have to cross w with?]

Determine whether u, v, and w lie in the same plane.

12.6 PLANES IN 3-SPACE

Equation of plane: Concept: normal vector and displacement vector (from fixed point (x₀,y₀,z₀) to arbitrary point (x,y,z) ) are perpendicular:

Vector form using dot product of a normal vector and a displacement vector:

n dot (r-r₀) = 0

Same thing in component form:

<a,b,c> dot (x-x₀, y-y₀, z-z₀> = 0

Take dot product to get point-normal form:

a(x-x0) + b(y-y0) + c(z-z0) = 0

Clear paren and collect like terms to get general form:

ax + by +cz + d = 0, where a normal vector is n = <a,b,c>

Example ...

Angle between planes = angle between normals

Find cos q where q is the angle between n₁ and n₂

Example ...

Distance between point P (x₀,y₀,z₀) and plane ax+by+cz+d = 0

Concept: pick any point Q (x₁,y₁,z₁) in the plane.

The desired distance is the magnitude of the projection of the vector PQ (or QP) onto n

Vector formula: |QP dot n| / norm(n)

Scalar formula:

Where does formula come from?

Work out vector formula:

QP dot n = <x₀-x₁, y₀-y₁, z₀-z₁> dot (<a,b,c>

= ax₀ + by₀ + cz₀ - (ax₁ + by₁ + cz₁)

= ax₀ + by₀ + cz₀ - (-d) (note ax₁+by₁+cz₁+d = 0 because Q is in the plane)

Example ...

Alternate applications

Distance between parallel planes

The points P and Q are any two points, P in the first plane, Q in the other.

Distance between skew lines

Pick any two points P and Q on the two lines

Find any two vectors v₁ and v₂ parallel to the two lines

The normal vector (to parallel planes containing the two lines) is v₁ cross v₂

12.7 QUADRIC SURFACES

Visualization (typical): collection of traces of curves parallel to the coordinate planes

Maple (for a function): plot3d(expression in x and y, x=a..b, y=c..d);

Quadric surfaces: Extensions of conic sections to 3D

For quadric surfaces, traces are conic sections

Illustration/summary on pg 841

Equations of quadric surfaces: second degree equations in 3 variables

See general form on page 840

Your tasks:

Identify which kind of quadric section, given an equation

Sketch each type of quadric section (rough), given an equation

Easiest: ellipsoid: Example like #1

See traces in coordinate planes

Elliptic paraboloid: Ex like Ex 5

See traces in coord planes and trace at z = constant (for z=f(x,y)

Hyperboloids

One sheet: Ex: Ex 2

Eqn: note one "-" signs

Sketch hyperbola (both sides) and parallel ellipses linking the two sides

Two sheets: Ex 3

Eqn: note two "-" signs

Sketch hyperbola (both sides) and parallel ellipses keeping the two sides separate

Cones Compare/contrast eqn with that of a hyperboloid: " = 0 " rather than " = 1 "

Hyperbolic paraboloid

Hardest to sketch

Example: z = x₂ - y₂

First, sketch parabola opening upward in xz=plane (z = x², with y=0)

Next, sketch parabolas of the same shape in parallel planes (traces for y = ±k)

Shift them down (because of the -y²)

Extend the tops of these parabolas upward to the same level as the first one

Link the vertices of the 3 parabolas with a parabola opening downward (z = -y²)

Link the tops of the 3 parabolas with the two sides of a hyperbola (x₂ - y₂ = k)

Sketch additional traces parallel to the original parabolas:

(sketch only the portions of each curve that are visible; don't sketch what is hidden)

Equations of Surfaces of Revolution

Earlier lessons in calculus cover

Volumes of ...

Surface areas of ...

Now we cover the equations themselves: [you should complete the table yourself]

For an equation in:	Revolved about	Replace	With
x and y	x-axis [must be one of the original variables]	y²[the second original variable]	y² + z²[both the second original variable and the third variable]
x and y	y-axis	x²	x² + z²
x and z	x-axis	z²	z² + y²
x and z	z-axis	x²	x²+ y²
y and z	y-axis	?	?
y and z	z-axis	?	?

Ex: y = x² about y-axis: Replace x² by x² + z²; we get y = x² + z², a paraboloid

Ex: y = x² about x-axis: To replace y² we first square both sides, getting y² = x⁴. Then we replace y² by y² + z², resulting in the equation y² + z² = x⁴. To graph in rectangular coordinates using Maple, solve for z:

12.8..Cylindrical and Spherical Coordinates

Converting points from one system to another

Text examples 1,3

Converting equations from one system to another

Text example 3

Equations with a single variable, in each system

x or y or z = constant

Cyl: r or q or z = constant

Sph: rho or q or φ = constant

Converting longitudes and latitudes in navigation to spherical coords

E Long: q = Long

W Long: q = 360° - Long

N Lat: φ = 90°- Lat

S Lat: φ = 90° + Lat

rho = radius of earth, about 4000 mi.

App: convert spherical coords to rectangular coords, find angle between vectors from center of earth, use s=rq to find great circle dist between two cities.

Equations of surfaces of revolution

About z-axis, in cylindrical or spherical coords (covered by text: use conversions to ...)

About any coord axis, in rect coords: Note the pattern in the examples below:

If eqn is in x and y, and revolving about y-axis

replace x (the other var in the eqn) by sqrt(x²+z²) thus introducing the other variable

clear the radical if possible

If eqn is in x and z, and revolving about x-axis

replace z (the other var in the eqn) by sqrt(z²+y²) thus introducing the other variable

clear the radical if possible

Return to: Merced College; Don Power Updated 01/29/07 by Don Power