Download Wiley

Chapter 9-Vectors
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.1 Vectors in the Plane
Chapter 9-Vectors
INSERT FIGURE 9-1-1
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.1 Vectors in the Plane
Chapter 9-Vectors
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.1 Vectors in the Plane
Chapter 9-Vectors
Vectors
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.1 Vectors in the Plane
Chapter 9-Vectors
Vector Algebra
DEFINTION: The sum v+w of two vectors
v=<v1,v2> and w=<w1,w2> is formed by adding the
vectors componentwise:
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.1 Vectors in the Plane
Chapter 9-Vectors
Vector Algebra
EXAMPLE: Add the vectors v = <−3, 9> and
w = <1, 8>.
DEFINITION: The zero vector 0 is the vector
both of whose components are 0.
DEFINITION: If v = <v1, v2> is a vector and l
is a real number then we define the scalar
multiplication of v by l to
be
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.1 Vectors in the Plane
Chapter 9-Vectors
The Length (or Magnitude) of a
Vector
THEOREM: If v is a vector and l is a scalar, then
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.1 Vectors in the Plane
Chapter 9-Vectors
Unit Vectors and Directions
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.1 Vectors in the Plane
Chapter 9-Vectors
Unit Vectors and Directions
DEFINITION: Let v and w be nonzero vectors.
We say that v and w have the same direction if
dir(v) =dir(w). We say that v and w are opposite
in direction if dir(v) = −dir(w). We say that v and
w are parallel if either (i) v and w have the
same direction or (ii) v and w are opposite in
direction. Although the zero vector 0 does not
have a direction, it is conventional to say that 0
is parallel to every vector.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.1 Vectors in the Plane
Chapter 9-Vectors
Unit Vectors and Directions
THEOREM: Vectors v and w are parallel if and only
if at least one of the following two equations
holds: (i) v = 0 or (ii) w = lv for some scalar l.
Moreover, if v and w are both nonzero and w = l
v, then v and w have the same direction if 0 < l
and opposite directions if l < 0.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.1 Vectors in the Plane
Chapter 9-Vectors
Unit Vectors and Directions
EXAMPLE: For what value of a are the vectors
<a,−1> and <3, 4> parallel?
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.1 Vectors in the Plane
Chapter 9-Vectors
An Application to Physics
EXAMPLE: Two workers are each pulling on a rope
attached to a dead tree stump. One pulls in the
northerly direction with a force of 100 pounds and
the other in the easterly direction with a force of
75 pounds. Compute the resultant force that is
applied to the tree stump.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.1 Vectors in the Plane
Chapter 9-Vectors
The Special Unit Vectors i and j
i=<1,0> and j=<0,1>
Suppose that v=<3,-5> and w=<2,4>. Express v,
w, and v+w in terms of i and j.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.1 Vectors in the Plane
Chapter 9-Vectors
The Triangle Inequality
EXAMPLE: Verify the Triangle Inequality for the
vectors v = <−3, 4> and w = <8, 6>.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.1 Vectors in the Plane
Chapter 9-Vectors
Quick Quiz
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.2 Vectors in ThreeDimensional Space
Chapter 9-Vectors
EXAMPLE: Sketch the points (3,2,5),
(2,3,-3), and (-1,-2,1).
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.2 Vectors in Three-Dimensional Space
Chapter 9-Vectors
Distance
THEOREM: The distance d(P,Q) between
points P=(p1, p2, p3) and Q=(q1, q2, q3) is given
by
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.2 Vectors in Three-Dimensional Space
Chapter 9-Vectors
Distance
EXAMPLE: Determine what set of points is
described by the equation
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.2 Vectors in Three-Dimensional Space
Chapter 9-Vectors
Distance
DEFINITION: Let P0 = (x0, y0, z0) be a point in space
and let r be a positive number. The set
is the set of all points inside the sphere
This set is called the open ball with center P0 and
radius r.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.2 Vectors in Three-Dimensional Space
Chapter 9-Vectors
Vectors in Space
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.2 Vectors in Three-Dimensional Space
Chapter 9-Vectors
Vector Operations
EXAMPLE: Suppose v=<3,0,1> and w=<0,4,2>.
Calculate v+w, and sketch the three vectors.
EXAMPLE: Suppose v=<2,-1,1>. Calculate 3v and
-4v.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.2 Vectors in Three-Dimensional Space
Chapter 9-Vectors
The Length of a Vector
If v=<v1,v2,v3> is a vector, then the length or
magnitude of v is defined to be
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.2 Vectors in Three-Dimensional Space
Chapter 9-Vectors
Unit Vectors and Directions
EXAMPLE: Is there a value of r for which
u=<-1/3,2/3,r> is a unit vector?
EXAMPLE: Suppose that v=<4,3,1> and w=<2,b,c>.
Are there values of b and c for which v and w are
parallel?
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.2 Vectors in Three-Dimensional Space
Chapter 9-Vectors
Quick Quiz
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.3 The Dot Product and
Applications
Chapter 9-Vectors
The Algebraic Definition of the Dot Product
DEFINITION: The dot product vw of two vectors v and
w is the sum of the products of corresponding entries
of v and w.
EXAMPLE: Let u = <2, 3,−1>, v = <4, 6,−2>, and w =
<−2,−1,−7>. Calculate the dot products u · v, u · w,
and v · w.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.3 The Dot Product and Applications
Chapter 9-Vectors
The Algebraic Definition of the Dot Product
THEOREM: Suppose that u, v, and w are vectors and
that l is a scalar. The dot product satisfies the
following elementary properties:
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.3 The Dot Product and Applications
Chapter 9-Vectors
The Algebraic Definition of the Dot Product
EXAMPLE: Let u=<3,2> and v=<4,-5>.
Calculate (u v)u+(v u)v
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.3 The Dot Product and Applications
Chapter 9-Vectors
A Geometric Formula for the Dot Product
THEOREM: Let v and w be nonzero vectors. Then
the angle q between v and w satisfies the
equation
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.3 The Dot Product and Applications
Chapter 9-Vectors
A Geometric Formula for the Dot Product
EXAMPLE: Calculate the angle between the two
vectors v=<2,2,4>and w=<2,-1,1>.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.3 The Dot Product and Applications
Chapter 9-Vectors
A Geometric Formula for the Dot Product
Cauchy-Schwarz Inequality:
EXAMPLE: Verify that the two vectors v = <2, 2, 4>
and w = <12, 13, 24> satisfy the Cauchy-Schwarz
Inequality.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.3 The Dot Product and Applications
Chapter 9-Vectors
A Geometric Formula for the Dot Product
DEFINTION: Let q be the angle between nonzero
vectors v and w. If q = p/2 then we say that the
vectors v and w are orthogonal or mutually
perpendicular.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.3 The Dot Product and Applications
Chapter 9-Vectors
A Geometric Formula for the Dot Product
THEOREM: Let v and w be any vectors. Then:
a) v and w are orthogonal if and only if v · w = 0.
b) v and w are parallel if and only if
c) If v and w are nonzero, and if q is the angle between
them, then v and w are parallel if and only if q = 0
or q = p. In this case, v and w have the same direction if
q = 0 and opposite directions if q = p.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.3 The Dot Product and Applications
Chapter 9-Vectors
A Geometric Formula for the Dot Product
EXAMPLE: Consider the vectors u = <2, 3,−1>, v = <4,
6,−2>, and w = <−2,−1,−7>. Are any of these vectors
orthogonal? Parallel?
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.3 The Dot Product and Applications
Chapter 9-Vectors
Projection
THEOREM: If v and w are nonzero vectors then
the projection Pw(v) of v onto w is given by
The length of Pw(v) is given by
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.3 The Dot Product and Applications
Chapter 9-Vectors
Projection
EXAMPLE: Let v = <2, 1,−1> and w = <1,−2, 2>.
Calculate the projection of v onto w, the
projection of w onto v, and calculate the lengths
of these projections.
Also calculate the component of v in the
direction of w and the component of w in the
direction of v.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.3 The Dot Product and Applications
Chapter 9-Vectors
Projection and the Standard Basis Vectors
EXAMPLE: Let v = <2,-6,12>. Calculate Pi(v), Pj(v),
and Pk(v) and express v as a linear combination of
these projections.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.3 The Dot Product and Applications
Chapter 9-Vectors
Direction Cosines and Direction Angles
EXAMPLE: Calculate the direction cosines and
direction angels for the vector
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.3 The Dot Product and Applications
Chapter 9-Vectors
Applications
EXAMPLE: A tow truck pulls a disabled vehicle a
total of 20,000 feet. In order to keep the vehicle
in motion, the truck must apply a constant force
of 3, 000 pounds. The hitch is set up so that the
force is exerted at an angle of 30 with the
horizontal. How much work is performed?
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.3 The Dot Product and Applications
Chapter 9-Vectors
Quick Quiz
1. Calculate <1, 2,−1> · <3, 4, 2>.
2. Use the arccosine to express the angle
between <6, 3, 2> and <4,−7, 4>.
3. For what value of a are <1, a,−1> and <3, 4, a>
perpendicular?
4. Calculate the projection of <3, 1,−1> onto
<8, 4, 1>.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.4 The Cross Product and
Triple Product
Chapter 9-Vectors
The Cross Product of Two Spatial Vectors
DEFINTION: If v = <v1, v2, v3> and w = <w1,w2,w3>,
then we define their cross product v × w to be
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.4 The Cross Product and Triple Product
Chapter 9-Vectors
The Cross Product of Two Spatial Vectors
THEOREM: If v and w are vectors, then v × w is
orthogonal to both v and w.
EXAMPLE: Let v = <2,−1, 3> and w = <5, 4,−6>.
Calculate v × w. Verify that v × w is orthogonal to
both v and w.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.4 The Cross Product and Triple Product
Chapter 9-Vectors
The Relationship between Cross Products and
Determinants
If v = <v1, v2, v3> and w = <w1,w2,w3>, then
EXAMPLE: Use a determinant to calculate the cross
product of v = <2,−1, 6> and w = <−3, 4, 1>.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.4 The Cross Product and Triple Product
Chapter 9-Vectors
Algebraic Properties of the Cross Product
If u, v, and w are vectors and l and m are scalars,
then
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.4 The Cross Product and Triple Product
Chapter 9-Vectors
Algebraic Properties of the Cross Product
THEOREM: If v is any vector, then v×v = 0. More
generally, if u and v are parallel vectors then
u×v = 0.
EXAMPLE: Give an example to show that the cross
product does not satisfy a cancellation property.
Give an example to show that the cross product
does not satisfy the associative property.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.4 The Cross Product and Triple Product
Chapter 9-Vectors
A Geometric Understanding of the Cross
Product
EXAMPLE: Find the standard unit normal for the
pairs (i, j) and (j, k) and (k, i). Find also the
standard unit normal for the pairs (j, i) and (k, j)
and (i, k).
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.4 The Cross Product and Triple Product
Chapter 9-Vectors
A Geometric Understanding of the Cross Product
THEOREM: Let v and w be vectors. Then
a)
b) If v and w are nonzero, then
where q [0,p] denotes the angle between v and w;
c) v and w are parallel if and only if v × w = 0;
d) If v and w are not parallel, then v × w points in
the direction of the standard unit normal for the
pair (v,w).
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.4 The Cross Product and Triple Product
Chapter 9-Vectors
A Geometric Understanding of the Cross Product
EXAMPLE: Let v = <1,−3, 2> and w = <1,−1, 4>.
What is the standard unit normal vector for
(v,w)?
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.4 The Cross Product and Triple Product
Chapter 9-Vectors
Cross Products and the Calculation of Area
THEOREM: Suppose that v and w are nonparallel
vectors. The area of the triangle determined by v
and w is
The area of the parallelogram determined by v
and w is
EXAMPLE: Find the area of the parallelogram
determined by the vectors v=<-2,1,3> and
w=<1,0,4>.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.4 The Cross Product and Triple Product
Chapter 9-Vectors
The Triple Scalar Product
DEFINITION: If u, v, and w are given vectors, then
we define their triple scalar product to be the
number (u×v) · w.
EXAMPLE: Calculate the triple scalar product of
u = <2,−1, 4>, v = <7, 2, 3>, and w = <−1, 1, 2> in
two different ways.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.4 The Cross Product and Triple Product
Chapter 9-Vectors
The Triple Scalar Product
THEOREM: The triple scalar product of
u = <u1, u2, u3>, v = <v1, v2, v3>, and
w = <w1,w2,w3> is given by the formula
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.4 The Cross Product and Triple Product
Chapter 9-Vectors
The Triple Scalar Product
EXAMPLE: Use the determinant to calculate
the volume of the parallelepiped determined
by the vectors <−3, 2, 5>, <1, 0, 3>, and
<3,−1,−2>.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.4 The Cross Product and Triple Product
Chapter 9-Vectors
The Triple Scalar Product
THEOREM: Three vectors u ,v, and w are
coplanar if and only if u · (v × w) = 0.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.4 The Cross Product and Triple Product
Chapter 9-Vectors
Triple Vector Products
DEFINITION: If u, v, and w are given spatial
vectors, then each of the vectors u × (v × w)
and (u × v) × w is said to be a triple vector
product of u, v, and w.
EXAMPLE: Let v and w be perpendicular
spatial vectors. Show that
and
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.4 The Cross Product and Triple Product
Chapter 9-Vectors
Quick Quiz
1. Calculate <2, 1, 2> × <1,−2,−1>.
2. Find the area of the parallelogram determined by <2,
1,−2> and <1, 1, 0>.
3. Find the standard unit normal vector for the ordered
pair (<2, 1,−2>, <1, 1, 0>) .
4. True or false: a) v × w = w × v b) u × (v × w) = (u × v) × w
c) u · v × w = u × v · w?
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.5 Lines and Planes in Space
Chapter 9-Vectors
Cartesian Equations of Planes in Space
THEOREM: Let V be a plane in space. Suppose
that n=<A,B,C> is a normal vector for V and
that P0=(x0,y0,z0) is a point on V. Then
is a Cartesian equation for V.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.5 Lines and Planes in Space
Chapter 9-Vectors
Cartesian Equations of Planes in Space
THEOREM: Suppose at least one of the coefficients A, B, C is
nonzero. Then the solution set of the equation
A(x−x0)+B(y−y0)+C(z−z0) = 0
is the plane that has <A,B,C> as a normal vector and passes
through the point (x0, y0, z0). The solution set of the
equation Ax + By + Cz = D is a plane that has <A,B,C> as a
normal vector.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.5 Lines and Planes in Space
Chapter 9-Vectors
Cartesian Equations of Planes in Space
EXAMPLE: Find an equation for the plane V passing through
the points P = (2,−1, 4), Q = (3, 1, 2), and R = (6, 0, 5).
EXAMPLE: Find the angle between the plane with Cartesian
equation x − y − z = 7 and the plane with Cartesian equation
−x + y − 3z = 6.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.5 Lines and Planes in Space
Chapter 9-Vectors
Parametric Equations of Planes in Space
THEOREM: If P0 = (x0, y0, z0) is a point on a plane V and if u =
<u1, u2, u3> and v = <v1, v2, v3> are any two nonparallel
vectors that are perpendicular to a normal vector for V, then
V consists precisely of those points (x, y, z) with coordinates
that satisfy the vector equation
When written coordinatewise, equation above yields
parametric equations for V:
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.5 Lines and Planes in Space
Chapter 9-Vectors
Parametric Equations of Planes in Space
EXAMPLE: Find parametric equations for the
plane V whose Cartesian equation is
3x − y + 2z = 7.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.5 Lines and Planes in Space
Chapter 9-Vectors
Parametric Equations of Lines in Space
THEOREM: The line in space that passes through the
point P0 = (x0, y0, z0) and is parallel to the vector m = <a,
b, c> has equation
Here P = (x, y, z) is a variable point on the line. In
coordinates the equation may be written as three
parametric equations:
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.5 Lines and Planes in Space
Chapter 9-Vectors
Parametric Equations of Lines in Space
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.5 Lines and Planes in Space
Chapter 9-Vectors
Cartesian Equations of Line in Space
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.5 Lines and Planes in Space
Chapter 9-Vectors
Cartesian Equations of Line in Space
EXAMPLE: Find parametric equations of the line
of intersection of the two planes
x − 2y + z = 4 and 2x + y − z = 3.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.5 Lines and Planes in Space
Chapter 9-Vectors
Cartesian Equations of Line in Space
EXAMPLE: Find parametric equations of the line
of intersection of the two planes
x − 2y + z = 4 and 2x + y − z = 3.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.5 Lines and Planes in Space
Chapter 9-Vectors
Calculating Distance
THEOREM: Suppose that P = (x0, y0, z0) is a point
and that V is a plane. Let n = <A,B,C> be a normal
vector for V and let Q = (x1, y1, z1) be any point on
V. The distance between P and V is equal to
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.5 Lines and Planes in Space
Chapter 9-Vectors
Calculating Distance
EXAMPLE: Find the distance between the point
P = (3,−8, 3) and the plane V whose Cartesian
equation is 2x + y − 2z = 10.
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved
9.5 Lines and Planes in Space
Chapter 9-Vectors
Quick Quiz
Calculus, 2ed, by Blank & Krantz, Copyright 2011
by John Wiley & Sons, Inc, All Rights Reserved