Chapter 9-Vectors Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All...

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Chapter 9-Vectors

Chapter 9-Vectors9.1 Vectors in the Plane

INSERT FIGURE 9-1-1

Vectors

DEFINTION: The sum v+w of two vectors v=<v1,v2> and w=<w1,w2> is formed by adding the vectors componentwise:

Vector Algebra

EXAMPLE: Add the vectors v = <−3, 9> and w = <1, 8>.

Vector Algebra

DEFINITION: The zero vector 0 is the vector both of whose components are 0.

DEFINITION: If v = <v1, v2> is a vector and is a real number then we define the scalar multiplication of v by tobe

The Length (or Magnitude) of a Vector

THEOREM: If v is a vector and is a scalar, then

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.

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 = v for some scalar . Moreover, if v and w are both nonzero and w = v, then v and w have the same direction if 0 < and opposite directions if < 0.

EXAMPLE: For what value of a are the vectors <a,−1> and <3, 4> parallel?

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.

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.

The Triangle Inequality

EXAMPLE: Verify the Triangle Inequality for the vectors v = <−3, 4> and w = <8, 6>.

Quick Quiz

Chapter 9-Vectors9.2 Vectors in Three-Dimensional Space

EXAMPLE: Sketch the points (3,2,5), (2,3,-3), and (-1,-2,1).

Distance

THEOREM: The distance d(P,Q) between points P=(p1, p2, p3) and Q=(q1, q2, q3) is given by

Distance

EXAMPLE: Determine what set of points is described by the equation

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.

Vectors in Space

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.

The Length of a Vector

If v=<v1,v2,v3> is a vector, then the length or magnitude of v is defined to be

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?

Quick Quiz

Chapter 9-Vectors9.3 The Dot Product and Applications

The Algebraic Definition of the Dot Product

DEFINITION: The dot product vw 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.

THEOREM: Suppose that u, v, and w are vectors and that is a scalar. The dot product satisfies the following elementary properties:

EXAMPLE: Let u=<3,2> and v=<4,-5>. Calculate (u v)u+(v u)v

A Geometric Formula for the Dot Product

THEOREM: Let v and w be nonzero vectors. Then the angle between v and w satisfies the equation

EXAMPLE: Calculate the angle between the two vectors v=<2,2,4>and w=<2,-1,1>.

Cauchy-Schwarz Inequality:

EXAMPLE: Verify that the two vectors v = <2, 2, 4> and w = <12, 13, 24> satisfy the Cauchy-Schwarz Inequality.

DEFINTION: Let be the angle between nonzero vectors v and w. If = /2 then we say that the vectors v and w are orthogonal or mutually perpendicular.

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 is the angle between them, then v and w are parallel if and only if = 0or = . In this case, v and w have the same direction if = 0 and opposite directions if = .

EXAMPLE: Consider the vectors u = <2, 3,−1>, v = <4, 6,−2>, and w = <−2,−1,−7>. Are any of these vectors orthogonal? Parallel?

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

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.

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.

Direction Cosines and Direction Angles

EXAMPLE: Calculate the direction cosines and direction angels for the vector

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?

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>.

Chapter 9-Vectors9.4 The Cross Product and Triple Product

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

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.

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>.

Algebraic Properties of the Cross Product

If u, v, and w are vectors and and are scalars, then

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.

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).

THEOREM: Let v and w be vectors. Thena) b) If v and w are nonzero, thenwhere [0,] 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).

EXAMPLE: Let v = <1,−3, 2> and w = <1,−1, 4>. What is the standard unit normal vector for (v,w)?

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>.

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.

THEOREM: The triple scalar product ofu = <u1, u2, u3>, v = <v1, v2, v3>, and w = <w1,w2,w3> is given by the formula

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>.

THEOREM: Three vectors u ,v, and w are coplanar if and only if u · (v × w) = 0.

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

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?

Chapter 9-Vectors9.5 Lines and Planes in Space

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.

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.

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.

Parametric Equations of Planes in SpaceTHEOREM: 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:

Parametric Equations of Planes in Space

EXAMPLE: Find parametric equations for the plane V whose Cartesian equation is 3x − y + 2z = 7.

Parametric Equations of Lines in SpaceTHEOREM: 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:

Parametric Equations of Lines in Space

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.

EXAMPLE: Find parametric equations of the line of intersection of the two planes

x − 2y + z = 4 and 2x + y − z = 3.

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

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.

Quick Quiz

Chapter 9-Vectors Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All...

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