12.2 Vectors

Ulrich Hoensch

Wednesday, August 26, 2009

Definition of Vectors

A vector is a directed line segment or an “arrow.” It has an initialpoint and a terminal point.

Definition of Vectors

I If A = (x1, y1) is the initial point and of the vector v andB = (x2, y2) is the terminal point of the vector v, then

v =−→AB = 〈x2 − x1, y2 − y1〉.

I If A = (x1, y1, z1) is the initial point and of the vector v andB = (x2, y2, z2) is the terminal point of the vector v, then

v =−→AB = 〈x2 − x1, y2 − y1, z2 − z1〉.

In applications, displacement, velocity, momentum,acceleration, and force are modeled using vectors.

Length (Magnitude) of a Vector

Definition 12.3

I If v = 〈v1, v2〉 is a vector in R2, then the length ormagnitude of v is

|v| =√

v21 + v2

2 .

I If v = 〈v1, v2, v3〉 is a vector in R3, then the length ormagnitude of v is

|v| =√

v21 + v2

2 + v23 .

Note that the length of a vector is simply the distance from itsinitial point to its terminal point.

Polar Coordinates

In 2-space, a vector can also be represented in polar coordinates(instead of using Cartesian coordinates).

I If v = 〈v1, v2〉, then r =√

v21 + v2

2 , and tan θ = v2/v1.

I If v = 〈v1, v2〉 has polar coordinates r and θ, then v1 = r cos θand v2 = r sin θ.

Vector Algebra

Definition 12.4

I Let u = 〈u1, u2〉 and v = 〈v1, v2〉 be vectors in R2, and let kbe a real number (a scalar). Then,

I the sum of u and v is the vector u + v = 〈u1 + v1, u2 + v2〉;I the scalar product of u and k ∈ R is the vector

ku = 〈ku1, ku2〉.

I Let u = 〈u1, u2, u3〉 and v = 〈v1, v2, v3〉 be vectors in R3, andlet k be a real number (a scalar). Then,

I the sum of u and v is the vectoru + v = 〈u1 + v1, u2 + v2, u3 + v3〉;

I the scalar product of u and k ∈ R is the vectorku = 〈ku1, ku2, ku3〉.

Vector Algebra

Properties

I |ku| = |k ||u|;I if k > 0, then ku is the vector u, stretched by factor k ;

I if k < 0, then ku is the vector u, stretched by factor k , butpointing in the opposite direction;

I additional properties on p. 841.

Vector Algebra

We define the vector −u to be (−1)u; also, the difference of twovectors u− v is defined as u + (−v).

Unit Vectors

Definition 12.5

A unit vector is any vector u with |u| = 1.

I The standard (Cartesian) unit vectors in R2 are i = 〈1, 0〉and j = 〈0, 1〉. Any vector v = 〈v1, v2〉 can thus be written asv = v1i + v2j.

I The standard (Cartesian) unit vectors in R3 are i = 〈1, 0, 0〉,j = 〈0, 1, 0〉 and k = 〈0, 0, 1〉. Any vector v = 〈v1, v2, v3〉 canthus be written as v = v1i + v2j + v3k.

If v is any non-zero vector, then its associated unit vector, ordirection vector is u = (1/|v|)v. Consequently, v can be writtenin its magnitude-direction form as v = |v|u.

Practice Problems for Section 12.2

p.844-846: 1, 7, 13, 15, 21, 25, 31, 33, 41, 43, 47.