1. 5.4 Vectors terminal point initial point Q v S w P R
A geometric vector in 2-dimensional space is a directed line segment with initial point P and terminal point Q. A vector such as this would be denoted as
v = PQ
The magnitude of this vector is the length of the line segment from P to Q. This is represented by ||v||.
A vector with P = Q is called a zero vector because the magnitude = 0.
Vectors with the same magnitude and same direction are equal.
v =PQ = w = RS
Notice that even
though v and w have
different initial points
and different terminal
points, they can still
be equal because they
have the same
magnitudes and
directions.
terminal point Q P
initial point Q v S w P θ R θ

2. Vector addition is commutative, which means v + w = w+v
Adding Vectors
You can add vectors v and w be repositioning them so that the terminal point of v coincides with the initial point of w. This results in a new vector, v+w. w v v
v+w w
Notice that the new vector v+w has the same initial point as v and the same terminal point as w (after they’ve been repositioned)
Vector addition is commutative,
which means v + w = w+v
Vector addition is also associative,
which means u+(v+w) = (u+v)+w w
w+v
v+w v v
u+v+w = u+(v+w) = (u+v)+w w
u+v u

3. Vector subtraction is done by adding opposite vectors.
That is, u – v = u + (-v), where –v is the vector having the same magnitude as v, but whose direction is opposite to v. -w -s v
Notice that
r – s = r +(-s)
v-w
r-s r
w – w = 0 w -w
Multiplying Vectors by Numbers.
When dealing with vectors, we refer to real numbers as scalars.
If a is a scalar and v is a vector, the scalar product of av is defined as follows:
1. If a>0, the product av is the vector whose magnitude is a times the magnitude of v and whose direction is the same as v.
If a<0, the product av is the vector whose magnitude is |a| times the magnitude of v and whose direction is opposite that of v.
If a=0 or if v=0, then αv = 0.
Properties:
0v =0
1v = v
-1v = -v
(a+b)v = av + bv
a(v+w) =av + aw
a(bv) =(ab)v -v w v 2w 2v

4. Example 1 v u w b) c) 2v 3w 2v+3w u 2v-w+u -w 2v-w NOWYOU TRY #13 2v
2 units up u
3 units up w
3 units right
2 units down
2 units right
1 unit right 2v 3w
2v+3w u
2v-w+u -w
2v-w
NOWYOU TRY #13 2v

5. A unit vector u is a vector u for which ||u|| = 1
i and j are special unit vectors that lie on the x-axis and y-axis, respectively. i = <1,0> and j = <0,1>
Any nonzero vector can be made into a unit vector by dividing itself by its magnitude.
This is a unit vector that as the same direction as v, but a magnitude of 1.
Position Vectors
An algebraic vector v is represented as v = , where a and b are scalars (real numbers) called the components of v. An algebraic vector may also be written in the form v = ai + bj. The a denotes the x-axis (horizontal) component and the b denotes the y-axis (vertical) component of v. (Notice this is an entirely different notation than the complex plane.)
A position vector is an algebraic vector whose initial point is the origin.
Theorem:
Suppose v is a vector with initial point P1 = (x1,y1), not necessarily the origin, and terminal point P2 = (x2,y2).
If v = P1P2,
then v is equal to the position vector
P2(5,6)
P1(1,3)
(4,3)=(5-1,6-3)

6. ai + bj vector form addition subtraction, and scalar multiplication.
Let v = a1i + b1j and w= a2i + b2j
You do #33 and #39
If the direction of a vector, v, is given by θ, which is the angle between x-axis (or unit vector, i) and v, then v can be expressed in
ai + bj form in term of its magnitude and direction as
v = ||v||[(cosθ)i + (sinθ)j] = ||v||(cosθ)i + ||v||(sinθ)j v θ 1
sin θ
cos θ
(cosθ, sinθ)
||v||

7. Applications
A velocity vector whose magnitude is an object’s speed and direction is the direction object is moving in.
A force vector is a vector whose magnitude is the amount of force acting on an object and the direction is the direction of the force (i.e., gravity is a force acting in the downward direction (-j)).
Example 6
A ball is thrown with an initial speed of 25 miles per hour in a direction that makes an angle of 30° with the positive x-axis. Express the velocity vector, v, in terms of I and j. What is the initial speed in the horizontal direction (i)? What is the initial speed in the vertical direction (j)?
The speed is the magnitude, so ||v|| = 25. θ = 30° .
The initial speed in the horizontal direction in the horizontal component of v. Let’s put v in ai+bj form in terms of θ .
v = 25[(cos 30°)i + (sin 30°)j]

8. F3 = Force of gravity down = <0, -1200>
Example 7 A box of supplies that weights 1200 pounds is suspended by two cables attached to the ceiling. What is the tension in the two cables?
30°
45° F1
150° F2
30°
45°
F1 and F2 are the forces of tension in the cables holding the box in equilibrium.
pounds
F3 = Force of gravity down = <0, -1200>
For static equilibrium, the sum of the force vectors must equal zero.
F1 + F2 + F3 = 0 = <0,0>
Therefore the sum of the i components will be zero, and the sum of the j components will be zero.
We must find the magnitude and direction of each of these vectors.
We know the magnitude and direction of F3. || F3 || = 1200, and the direction is straight down (270°). Now we must find the magnitude and directions of F1 and F2.
F1 =
F2 =

9. HOMEWORK
p.382 #7-13 odd, #25-49 odd, 61,65