Section 13.2 Vectors

SCALARS AND VECTORS A scalar quantity can be characterized by a single real number. Examples: area, volume, time, mass, temperature. A vector is a quantity involving both magnitude and direction and cannot be characterized completely by a single real number. Examples: force, velocity, acceleration.

VECTORS A directed line segment is used to represent a vector quantity. A directed line segment with initial point (tail), A, and terminal point (head), B, is denoted by A B

EQUIVALENT VECTORS Two vectors u and v are considered to be equivalent (or equal) if the have the same length and direction.

THE ZERO VECTOR The zero vector, denoted by 0, has length 0. It is the only vector with no specific direction.

DEFINITION OF VECTOR ADDITION If u and v are vectors positioned so that 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.

THE PARALLELOGRAM LAW Draw u and v so that they have the same initial point. Then draw an additional copy of each to form a parallelogram. Then u + v is the vector with this common tail and coinciding with the diagonal of the parallelogram that has u and v as sides.

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 of v if c < 0. If c = 0 or v = 0, then cv = 0.

COMMENTS ON SCALAR MULTIPLICATION Two nonzero vectors are called parallel if they are scalar multiples of each other. The vector −v = (−1)v has the same length as v but points in the opposite direction. We call it the negative of v.

DIFFERENCE OF TWO VECTORS By the difference u − v of two vectors we mean u − v = u + (−1)v

ALGEBRAIC VECTORS AND COMPONENTS Sometimes it is best to introduce a coordinate system and treat vectors algebraically. If we place the initial point of a vector a at the origin, then the terminal point of a has coordinates of the form (a 1, a 2 ) or (a 1, a 2, a 3 ) depending on whether our coordinate system is two- or three- dimensional. These coordinates are called the components of a and we write

REPRESENTATIONS AND POSITION VECTOR Any vector that is equivalent (equal) to a is called a representation of a. The vector a is called the position vector of the point P. Let a be the vector from the origin to the point P(a 1, a 2 ) or P(a 1, a 2, a 3 ). That is, or.

Given the points A(x 1, y 1, z 1 ) and B(x 2, y 2, z 2 ), the vector a with representation is FINDING AN ALGEBRAIC VECTOR

The magnitude or length of the vector v is the length of any of its representations and is denoted by the symbol |v| or ||v||. The length of the two-dimensional vector is The length of the three-dimensional vector is MAGNITUDE

SUM, DIFFERENCE, AND SCALAR MULTIPLICATION If, then NOTE: A similar result holds for two-dimensional vectors.

HIGHER DIMENSIONAL VECTORS We denote the set of all two-dimensional vectors by V 2 and the set of all three-dimensional vectors by V 3. An n-dimensional vector is an ordered n­ tuple where a 1, a 2,..., a n are real numbers that are called the components of a. The set of all n­ dimensional vectors is denoted by V n.

PROPERTIES OF VECTORS If a, b, and c are vectors in V n and c and d are scalars, then 1. a + b = b + a2. a + (b + c) = (a + b) + c 3. a + 0 = a4. a + (−a) = 0 5. c(a + b) = ca + cb6. (c + d)a = ca + da 7. (cd)a = c(da)8.1a = a

The vectors are called the standard basis vectors since any vector in V 3 can be written in terms of them as follows THE STANDARD BASIS VECTORS

UNIT VECTORS A unit vector is a vector whose length is 1. For instance, i, j, and k are all unit vectors. In general, if a ≠ 0, then the unit vector that has the same direction as a is

APPLICATIONS OF VECTORS 1.An object weighing pounds is held in place by two ropes that make angles of º and 39.22º, respectively, with the vertical. Find the magnitude of the force exerted on the object by each rope. 2.A ship is sailing due south at 20 miles per hour. A man walks west (i.e., at right angles to the side of the ship) across the deck at 3 miles per hour. What are the magnitude and direction of the his velocity relative to the surface of the water.