Vectors and the geometry of space

Vectors The term vector is used by scientists to indicate a quantity that involves both magnitude and direction, such as force, velocity, and acceleration... A vector is often represen

1 Vectors and the Geometry of Space

1.1 Three-Dimensional Coordinate Systems

In order to represent points in space, we first choose a fixed point O (the origin) and three

directed lines through that are perpendicular to each other, called the coordinate axes and

labeled the x-axis, y-axis, and z-axis

To determine the orientation of a three-dimensional coordinate system, in this text we use the

right-handed rule as illustrated in the following figure

Now if P is any point in space, let a be the (directed) distance from the yz-plane to P, let

b be the distance from the xz-plane to P and let c be the distance from the xy-plane to P We represent the point by the ordered triple (a, b, c) of real numbers and we call it the coordinates of

P

Distance Formula The distance d between two points ( , , )x y z1 1 1 and ( ,x y z2 2, 2) is

Example 1 Find the distance between the points (2, -1, 3) and (1, 0, -2)

Example 2 Find the midpoint of the line segment joining the points ( , , )x y z1 1 1 and ( ,x y z2 2, 2)

Example 3 Find an equation of a sphere with radius r and center ( ,x y z0 0, )0

(the standard equation of a sphere)

1.2 Vectors

The term vector is used by scientists to indicate a quantity that involves both magnitude

and direction, such as force, velocity, and acceleration

We denote a vector by printing a letter in boldface (v) or by putting an arrow above the letter

(𝑣⃗) A vector is often represented by a directed line segment, for instance 𝑃𝑄⃗⃗⃗⃗⃗⃗ has initial point

P and terminal point Q, and its length (or magnitude) is denoted by ||𝑃𝑄⃗⃗⃗⃗⃗⃗||

Vectors having the same length and the same direction (even though in a different

position) are equivalent (or equal) and we write u = v The zero vector, denoted by 0, has

length 0 It is the only vector with no specific direction

Components of a vector

If the initial point of a vector is the origin of a rectangular coordinate system, then its

terminal point has coordinates of the form (a1, a2) or (a1, a2, a3), depending on whether the

system is two- or three-dimensional These coordinates are called the components of the vector

Example: We can see in the following figure, all the geometric vectors are the representations

of the algebraic vector 𝒂 = 〈3,2〉:

In three dimensions, given the points A x y z( , , )1 1 1 and B x y z( ,2 2, 2), the vector a with

representation 𝐴𝐵⃗⃗⃗⃗⃗⃗ is

Properties of vectors in components form

• Equality of vectors:

Let u u u u1, 2, 3 and v  v v v1, 2, 3 be vectors in space

u = v if and only if u1 = v1, u2 = v2 and u3 = v3

• The magnitude or length of the vector v  v v v1, 2, 3 is the length of any of its

representations:

• The vector sum of u u u u1, 2, 3 and v  v v v1, 2, 3 is the vector:

u + v  u1v u1, 2v u2, 3v3

• To multiple a vector u u u u1, 2, 3 by a scalar c:

cu  cu cu cu1, 2, 3

and length of a scalar multiple:

||cu|| = |c| ||u||

If a is nonzero, the vector

u = = a

||a|| ||a||

has length 1 and the same direction as a, called unit vector in the direction of a

• The negative of u is the vector:

- u = (-1)u  u1, u2,u3

• The difference of u and v is:

u - v = u + (-v) u1v u1, 2v u2, 3v3

Properties of Vector Operations

Let u, v and w be vectors in the plane, and let c and d be scalars

The standard basis vectors

have length 1 and point in the directions of the positive x-, y-, and z-axes

If a a a a1, 2, 3 , then we can write

Thus any vector in space can be expressed in terms of i, j, and k

Similarly, in two dimensions, we can write

Where i = 〈 1,0〉 and j = 〈0,1〉

Vectors have many applications in physics and engineering One example is force A vector can be used to represent force, because force has both magnitude and direction If two or

more forces are acting on an object, then the resultant force on the object is the vector sum of

the vector forces

Example Two tugboats are pushing an ocean liner, as shown in the figure

Each boat is exerting a force of 400 pounds What is the resultant force on the ocean liner?

Solution

Using the figure, you can represent the forces exerted by the first and second tugboats as

The resultant force on the ocean liner is

So, the resultant force on the ocean liner is approximately 752 pounds in the direction of the

positive x-axis

1.3 The dot product

So far, we have two operations with vectors—vector addition and multiplication by a scalar—each of which yields another vector In this section, you will study a third vector operation, the dot product This product yields a scalar, rather than a vector

The dot product of u u u u, , and v  v v v, , is

This operation has some properties:

Let u, v, and w be vectors in the plane or in space and let c be a scalar

The angle between two nonzero vectors u and v is the angle 𝜃, 0 ≤ 𝜃 ≤ 𝜋, between their

respective standard position vectors

u.v = ||u|| ||v|| cos 𝜃

or, if u and v are nonzero,

The vectors u and v are orthogonal when u.v = 0

Direction Angles and Direction Cosines

The direction angles of a nonzero vector v are the angles α, β, and γ (in the interval

[0,π]) that v makes with the positive x-, y-, and z-axes, or with three unit vectors i, j, and k

The cosines of these direction angles, cos α, cos β, and cos γ, are called the direction cosines of the vector v

cos 𝛼 = 𝐯 ∙ 𝐢

‖𝐯‖‖𝐢‖=

𝑣1

‖𝐯‖

Similarly,

cos 𝛽 = 𝐯 ∙ 𝐣

‖𝐯‖‖𝐣‖=

𝑣2

‖𝐯‖

cos 𝛾 = 𝐯 ∙ 𝐤

‖𝐯‖‖𝐤‖=

𝑣3

‖𝐯‖

Consequently, any nonzero vector v in space has the normalized form

and because v/||v|| is a unit vector, it follows that

Example: Find the direction cosines and angles for the vector v = 2i + 3j + 4k

Projections

The vector projection of vector b onto vector a is denoted by proj a b

The scalar projection of b onto a (also called the component of b along a) is defined to be the

signed magnitude of the vector projection: compa b

Using the dot product, we get:

comp𝐚𝐛 = |𝐛| cos 𝜃 =𝐚 ∙ 𝐛

|𝐚| =

𝐚

|𝐚|∙ 𝐛

Notice that the vector projection is the scalar projection times the unit vector in the direction of

a

Example: Find the scalar projection and vector projection of b = 〈1,1,2〉 onto a = 〈−2,3,1〉 Work

The work W done by the constant force F acting along the line of motion of an object is

given by

When the constant force F is not directed along the line of motion, the work W done by the force

is

Thus, the work done by a constant force as its point of application moves along the vector is one

of the following

Example: To close a sliding door, a person pulls on a rope with a constant force of 50 pounds at

a constant angle of 60o as shown in the following figure Find the work done in moving the door

12 feet to its closed position

Solution Using a projection, you can calculate the work as follows

1.4 Cross Product

Many applications in physics, engineering, and geometry involve finding a vector inspace that is orthogonal to two given vectors In this section, you will study a product that will yield such a vector

Definition

If u u u u1, 2, 3 and v  v v v1, 2, 3 , then th cross product of u and v is the vector

u × v u v2 3u v u v3 2, 3 1u v u v1 3, 1 2u v2 1

A convenient way to calculate is to use the determinant form with cofactor expansion shown

below

Note the minus sign in front of the j-component

Example:

Algebraic Properties of the Cross Product

Geometric Properties of the Cross Product

The Triple Product

For vectors u, v, and w in space, the triple scalar product is:

Geometric Property of the Triple Scalar Product

The volume V of a parallelepiped with vectors u, v, and w as adjacent edges is V = |u ∙ (v × w)|

1.5 Lines and Planes

The parametric equations of a line parallel to the vector 𝐯 = 〈𝑎, 𝑏, 𝑐〉 and passing through

the point P(x0, y0, z0) is:

The symmetric equations of the line can be obtained by eliminating the parameter t, if the

direction numbers a, b, and c are all nonzero,

The standard equation of a plane containing the point (x0, y0, z0) and having normal vector

𝐧 = 〈𝑎, 𝑏, 𝑐〉 is

The linear equation (or general form of the equation) of a plane can be obtained by regrouping

terms,

ax + by + cz + d = 0

Example: Find the general equation of the plane containing the points (2, 1, 1), (0, 4, 1), (-2, 1, 4)

Example: Find the angle between the two planes

x – 2y + z = 0 and 2x + 3y – 2z = 0

Then find parametric equations of their line of intersection

Distances Between Points, Planes, and Lines

Thus the distance between the point (x1, y1, z1) and the plane having the linear equation ax + by + cz + d = 0 is

Example: Find the distance between the parallel planes 10x + 2y - 2z = 5 and 5x + y – z = 1

1.6 Cylinders and Quadratic Surfaces

Cylinders

A cylinder is a surface that consists of all lines (called rulings) that are parallel to a given line

and pass through a given plane curve

Example:

z = x2 is a parabolic cylinder

Quadric Surfaces

A quadric surface is the graph of a second-degree equation in three variables x, y, and z The

most general such equation is

There are six basic types of quadric surfaces: