Horizontal and vertical component of a two-dimensional
vector
v = ha; bi be a two-dimensional vector. This notation uses the Cartesian components of the vector: a is the horizontal component, b is the
vertical component.
Another way of describing this vec

Unit Vectors
A
q unit vector u = hu1 ; u2 i is a vector of length equal to 1: juj =
(u1 )2 + (u2 )2 = 1:
Given an arbitrary vector v = ha; bi, we can nd the corresponding unit
vector u (a vector with the same direction, but length 1) by dividing
the vecto

Review 12.2: Vectors in R2
Geometrically, a vector can be represented by an arrow with a specied
direction and length (magnitude). Notation: !
v , or v
Algebraically, a vector in R2 can be represented by an ordered pair of
real numbers: v = ha; bi :
The l

Vector components: a and b
a is the length of the projection of v onto the x-axis
b is the length of the projection of v onto the y-axis
Vector Operations
Addition
Algebraically, addition is done on components: u = hu1 ; u2 i, v =
hv1 ; v2 i, we have:

- parallelogram rule: place the two vectors such that they
have the same initial point, and sketch the diagonal of the
parallelogram determined by the two vectors, with the same
initial point.
Any vector v = hv1 ; v2 i in R2 can be written as the sum of

[Textbook 1.35) Compute the x and y-components of the vectors 21,131 5', and 15
shown.
(Textbook 1.39) For the vectors 2! and f? in the gure in Textbook 1.35 shown
above, use the method of components to nd the magnitude and direction of [a] the
vector s

Scalar multiplication
Algebraically: on components. For u = hu1 ; u2 i
2u = 2 hu1 ; u2 i = h2u1 ; 2u2 i
Geometrically: 2u is twice as long as u, same direction
q
q
2
2
j2uj = (2u1 ) + (2u2 ) = 2 (u1 )2 + (u2 )2 = 2 juj
For an arbitrary constant k:
ku =

Properties of Vector Operations
If u; v, and w are vectors in R2 , and c and d are scalars, then:
u + v = v + u (commutativity of vector addition)
u + (v + w) = (u + v) + w (associativity of vector addition)
u + 0 = u (the identity element for vector addi

Subtraction
Algebraically: on components. For vectors u = hu1 ; u2 i, v =
hv1 ; v2 i, we have:
u
v = hu1 ; u2 i
hv1 ; v2 i = hu1
v1 ; u2
v2 i ;
hence the result is a new vector.
Geometrically, it can be derived by using the two operations
dened above: wr

(Textbook 1.41) A disoriented physics professor drives 3.25 km north, then 4.75
km west, then 1.50 km south. Find the magnitude and direction of the resultant
dispiacement.
(Textbook 1.47) Write each vector shown in terms of the unit vectors f and .
[Te