ESSENTIAL CALCULUS CH10 Vectors and the geometry of space

In this Chapter: 10.1 Three-Dimensional Coordinate Systems 10.2 Vectors 10.3 The Dot Product 10.4 The Cross Product 10.5 Equations of Lines and Planes 10.6 Cylinders and Quadric Surfaces 10.7 Vector Functions and Space Curves 10.8 Arc Length and Curvature 10.9 Motion in Space: Velocity and Acceleration Review

Chapter 10, 10.1, P519

Chapter 10, 10.1, P520

Chapter 10, 10.1, P521

DISTANCE FORMULA IN THREE IMENSIONS The distance │ P 1 P 2 │ between the points P 1 (x 1,y 1,z 1 ) and P 2 (x 2,y 2,z 2 ) is

Chapter 10, 10.1, P522 EQUATION OF A SPHERE An equation of a sphere with center C( h, k, l) and radius r is In particular, if the center is the origin O, then an equation of the sphere is

Chapter 10, 10.2, P524

The term vector is used by scientists to indicate a quantity (such as displacement or velocity or force) that has both magnitude and direction. A vector is often represented by an arrow or a directed line segment. We denote a vector by printing a letter in boldface (v) or by putting an arrow above the letter (v).

Chapter 10, 10.2, P524 displacement vector v, shown in Figure 1, has initial point A (the tail) and terminal point B (the tip) and we indicate this by writing v= AB. Notice that the vector u= CD has the same length and the same direction as v even though it is in a different position. We say that u and v are equivalent (or equal) and we write u=v.

Chapter 10, 10.2, P524

Chapter 10, 10.2, P525 DEFINITION OF VECTOR ADDITION If u and v are vectors positioned so the initial point of v is at the terminal point of u, then the sum u + v is the vector from the initial point of u to the terminal point of v.

Chapter 10, 10.2, P525

DEFINITION OF SCALAR MULTIPLICATION If c is a scalar and v is a vector, then the scalar multiple cv is the vector whose length is │ c │ times the length of v and whose direction is the same as v if c>0 and is opposite to v if c<0. If c=0 or v=0, then cv=0.

Chapter 10, 10.2, P526

Notice that two nonzero vectors are parallel if they are scalar multiples of one another. In particular, the vector – v=(-1)v has the same length as v but points in the opposite direction. We call it the negative of v.

Chapter 10, 10.2, P526 By the difference u - v of two vectors we mean u - v= u + (-v)

Chapter 10, 10.2, P526

Chapter 10, 10.2, P527

1. Given the points A(x 1,y 1,z 1 ) and B(x 2,y 2,z 2 ), the vector a with representation AB is a=

Chapter 10, 10.2, P527

The length of the two-dimensional vector a= is The length of the three-dimensional vector a= is

Chapter 10, 10.2, P527 if a= and b=, then the sum is a + b= To add algebraic vectors we add their components. Similarly, to subtract vectors we subtract components. From the similar triangles in Figure 15 we see that the components of ca are ca 1 and ca 2. So to multiply a vector by a scalar we multiply each component by that scalar.

Chapter 10, 10.2, P528 If a= and b=, then Similarly, for three-dimensional vectors,

Chapter 10, 10.2, P528 We denote by V 2 the set of all two-dimensional vectors and by V 3 the set of all three- dimensional vectors. More generally, we will later need to consider the set V n of all n-dimensional vectors. An n-dimensional vector is an ordered n-tuple:

Chapter 10, 10.2, P528 PROPERTIES OF VECTORS If a, b, and c are vectors in V n and c and d are scalars, then 1. a + b=b + a 2. a + (b - c)=( a + b )+ c 3. a+0=a 4. a+(-a)=0 5. c(a + b)= ca + cb 6. (c + d) a= ca + da 7. (cd) a=c (da) 8. la=a

Chapter 10, 10.2, P529

Three vectors in V 3 play a special role. Let i= j= k= These vectors i,j, and k are called the standard basis vectors.

Chapter 10, 10.2, P529 If a=, then we can write Thus any vector in V 3 can be expressed in terms of i, j, and K.

Chapter 10, 10.2, P529 In two dimensions, we can write a= =a 1 i+a 2 j

Chapter 10, 10.3, P533 1.DEFINITION If a= and b=, then the dot product of a and b is the number a ‧ b given by

Chapter 10, 10.3, P PROPERTIES OF THE DOT PRODUCT If a, b, and c are vectors in V 3 and c is a scalar, then

Chapter 10, 10.3, P THEOREM If θ is the angle between the vectors a and b, then

Chapter 10, 10.3, P THEOREM If θ is the angle between the nonzero vectors a and b, then

Chapter 10, 10.3, P Two vectors a and b are orthogonal if and only if a ‧ b = 0.

Chapter 10, 10.3, P535 If S is the foot of the perpendicular from R to the line containing PQ, then the vector with representation PS is called the vector projection of b onto a and is denoted by prjo a b. (You can think of it as a shadow of b). The scalar projection of b onto a (also called the component of b along a) is defined to be numerically the length of the vector projection, which is the number │ b │ cosθ, where θ is the angle between a and b. (See Figure 4.) This is denoted by comp a b.

Chapter 10, 10.3, P535

Chapter 10, 10.3, P536

Scalar projection of b onto a: comp a b= Vector projection of b onto a: proj a b=

Chapter 10, 10.4, P DEFINITION If a= and b=, then the cross product of a and b is the vector

Chapter 10, 10.4, P539 A determinant of order 2 is defined by

Chapter 10, 10.4, P539 A determinant of order 3 can be defined in terms of second-order determinants as follows:

Chapter 10, 10.4, P540

Chapter 10, 10.4, P THEOREM The vector a ╳ b is orthogonal to both a and b.

Chapter 10, 10.4, P541

6. THEOREM If θ is the angle between a and b (so 0≤θ≤ ), then

Chapter 10, 10.4, P COROLLARY Two nonzero vectors a and b are parallel if and only if

Chapter 10, 10.4, P542 The length of the cross product a ╳ b is equal to the area of the parallelogram determined by a and b.

Chapter 10, 10.4, P542

Chapter 10, 10.4, P543

8. THEOREM If a, b, and c are vectors and c is a scalar, then

Chapter 10, 10.4, P544

11. The volume of the parallelepiped determined by the vectors a, b, and c is the magnitude of their scalar triple product:

Chapter 10, 10.5, P547

In 3-D space, a line L is determined if we know a point P o (x o, y o, z o) on L and the direction al vector V=. Let (x, y, z) be a point on L. Then (1)Pararnetric equations of L: 2.x=x o + at y=y o + Lt z=z o + ct (2)Symrnetric equations of L: 3.

Chapter 10, 10.5, P548 ▓ Figure 3 shows the line L in Example 1 and its relation to the given point and to the vector that gives its direction.

Chapter 10, 10.5, P549 ▓ Figure 4 shows the line L in Example 2 and the point P where it intersects the xy-plane.

Chapter 10, 10.5, P The line segment from r o to r 1 is given by the vector equation 0 ≤ t ≤1

Chapter 10, 10.5, P550 A plane in space is determined by a point P o (x o,y o,z o ) in the plane and a vector n that is orthogonal to the plane. This orthogonal vector n is called a normal vector.

Chapter 10, 10.5, P551 Let be P( x, y, z) be an arbitrary point in the plane, and let r 0 and r be the position vectors of P 0 and P. We have which can be rewritten as vector equation of the plane.

Chapter 10, 10.5, P551 We write n=,r=, and r o =. Then the vector equation (5) becomes Or Equation 7 is the scalar equation of the plane through P o (x 0,y 0,z o )with normal vector n=.

Chapter 10, 10.5, P551 By collecting terms in Equation 7, we can rewrite the equation of a plane as

Chapter 10, 10.5, P553 plane: ax + by + cz + d = 0

Chapter 10, 10.5, P553 Refers to fiy11. Thus the formula for D can be written as

Chapter 10, 10.6, P556 A quadric surface is the graph of a second- degree equation in three variables x, y, and z. The most general such equation is where A, B, C ‧‧‧ J are constants, but by translation and rotation it can be brought into one of the two standard forms or

Chapter 10, 10.6, P558 FIGURE 6 Vertical traces are parabolas; horizontal traces are hyperbolas. All traces are labeled with the value of k.

Chapter 10, 10.6, P558 FIGURE 7 Traces moved to their correct planes

Chapter 10, 10.6, P558 FIGURE 8 The surface z=y 2 -x 2 is a hyperbolic paraboloid.

Chapter 10, 10.6, P559 TABLE 1 Graphs of Quadric Surfaces All traces are ellipses. If a=b=c, the ellipsoid is a sphere.

Chapter 10, 10.6, P559 TABLE 1 Graphs of Quadric Surfaces Horizontal traces are ellipses. Vertical traces are parabolas. The variable raised to the first power indicates the axis of the paraboloid.

Chapter 10, 10.6, P559 TABLE 1 Graphs of Quadric Surfaces Horizontal traces are hyperbolas. Vertical traces are parabolas. The case where c<0 is illustrated.

Chapter 10, 10.6, P559 TABLE 1 Graphs of Quadric Surfaces Horizontal traces are ellipses. Vertical traces in the planes x=k and y=k are hyperbolas if k≠0 but are pairs of lines if k= 0.

TABLE 1 Graphs of Quadric Surfaces Horizontal traces are ellipses. Vertical traces are hyperbolas. The axis of symmetry corresponds to the variable whose coefficient is negative.

Chapter 10, 10.6, P559 TABLE 1 Graphs of Quadric Surfaces Horizontal traces in z=k are ellipses if k>c or k<-c. Vertical traces are hyperbolas. The two minus signs indicate two sheets.

Chapter 10, 10.7, P561 A vector-valued function, or vector function, is simply a function whose domain is a set of real numbers and whose range is a set of vectors.

Chapter 10, 10.7, P561 For every number t in the domain of r there is a unique vector in V 3 denoted by r(t). If f(t), g(t), and h(t) are the components of the vector r(t), then f, g, and h are real-valued functions called the component functions of r and we can write We use the letter t to denote the independent variable because it represents time in most applications of vector functions.

Chapter 10, 10.7, P561 If, then provided the limits of the component functions exist.

Chapter 10, 10.7, P562 FIGURE 1 C is traced out by the tip of a moving position vector r(t).

Chapter 10, 10.7, P563

Chapter 10, 10.7, P565 The derivative r ’ of a vector function r is defined in much the same way as for real valued functions:

Chapter 10, 10.7, P565 The vector r ’ (t) is called the tangent vector to the curve defined by r at the point P, provided that r ” (t) exists and r ” (t)≠0. The tangent line to C at P is defined to be the line through P parallel to the tangent vector r ’ (t). We will also have occasion to consider the unit tangent vector, which is

Chapter 10, 10.7, P565

Chapter 10, 10.7, P566 4, THEOREM If where f,g, and h are differentiable functions, then

Chapter 10, 10.7, P566

Chapter 10, 10.7, P567

FIGURE 13 The curve r(t)= is not smooth.

Chapter 10, 10.7, P568 5, THEOREM Suppose u and v are differentiable vector functions, c is a scalar, and f is a real- valued function. Then

Chapter 10, 10.7, P569

Chapter 10, 10.8, P572 The length of a plane curve with parametric equations x=f(t), y=g(t), a ≤ t ≤b, as the limit of lengths of inscribed polygons and, for the case where f ’ and g ’ are continuous, we arrived at the formula

Chapter 10, 10.8, P572 Suppose that the curve has the vector equation r(t)=, a ≤ t ≤b, then it can be show that its length is

Chapter 10, 10.8, P572 Notice that both of the arc length formulas (1) and (2) can be put into the more compact form

Chapter 10, 10.8, P572 FIGURE 1 The length of a space curve is the limit of lengths of inscribed polygons.

Chapter 10, 10.8, P573 ▓ Piecewise-smooth curves were introduced on page 567.

Chapter 10, 10.8, P573 Now we suppose that C is a piecewise-smooth curve given by a vector function r(t)= f(t)i + g(t)j+ h(t)k, a≤ t ≤ b, and C is traversed exactly once as increases from a to b. We define its arc length function s by Thus s(t) is the length of the part of C between r(a) and r(t). (See Figure 3.)

Chapter 10, 10.8, P574 FIGURE 4 Unit tangent vectors at equally spaced points on C

Chapter 10, 10.8, P DEFINITION The curvature of a curve is where T is the unit tangent vector. Namdy,

Chapter 10, 10.8, P574 We use the Chain Rule (Theorem , Formula 6) to write and But ds/dt= │ r ’ (t) │ from Equation 7, so

Chapter 10, 10.8, P THEOREM The curvature of the curve given by the vector function r is

Chapter 10, 10.8, P577 ▓ We can think of the normal vector as indicating the direction in which the curve is turning at each point.

Chapter 10, 10.8, P577 We can define the principal unit normal vector N(t) (or simply unit normal) as The vector B(t)=T(t) ╳ N(t) is called the binormal vector. It is perpendicular to both T and N and is also a unit vector. (See Figure 6.)

Chapter 10, 10.8, P578

Chapter 10, 10.9, P580

The speed of the particle at time t is the magnitude of he velocity vector, that is, │ v(t) │. rate of change of distance with respect to time