Notation:
(head)
B
a
(tail)
A
This vector can be written as:
a
a
AB
Often a vector is placed on a coordinate grid to help determine its length (magnitude), direction, etc... If its tail is at the origin it can be called OC, OC, or position vector of C
1
Magnitude ­ length of a vector (distance from tail to head).
notation: AB or just AB
2
Equal vectors
Two vectors are equal if they have the same direction and same magnitude.
ex. a
b
3
Negative vectors
A negative vector points in the opposite direction as its positive counterpart. It still has same magnitude.
a
­a
4
Drawing vectors ­ vectors should be drawn proportionally to each other.
a
1/2 a
2a
­4a
5
vector addition
there are two useful ways to geometrically interpret vector addition...
Head to Tail
(parallelogram)
b
Tail to Tail +b
a
resu
t a
ltan
b
a
a+
t
n
a
lt
resu
b
How do you decide which method to use? ­ depends on the given situation. 6
For the shape shown, find (or create) a vector which is equal to
B
C
a) AB + BC
b) AD + DB
A
D
c) BC + CD + DA
7
Subtraction of vectors ­ like vector addition, there are two useful ways to interpret vector subtraction...
Addition of the negative
Tail to Tail
show a­b
just connect tail to tail, arrow faces initial vector (a)
a + (­b)
.
8
More on Position Vectors.
Recall that a position vector assumes the tail is positioned at the origin. This is useful because then we can more accurately measure a vectors length and its direction.
This vector travels 8 units in the x­direction, and 3 units in the y­
direction.
a
Different ways to write this...
a = <8, 3>
a = 8i + 3j
( )
a = 8
3
The 8 and 3 are called the components of the vector. .
9
A vector that is not a position vector can still be described in the same manner.
b
b = <5, 3>
b = 5i + 3j
( )
b = 5
3
Even though it is not anchored to the origin, it is still made up of a 5i component and a 3j component.
10
Describe this vector...
c
11
Add vectors a and b.
a
b
What are the components of the resultant?
12
a
b
Subtract vectors a and b. What are the components of the resultant?
13
What about 3­D?
This works the same as 2­D vectors, it's just harder to draw and visualize. .
14
Z
7
a = <5, 8, 7>
a = 5i + 8j + 7k
a
8
Y
5
5
a = 8
7
( )
X
*The adding/subtracting works the same too. Again, it's just harder to draw and visualize.
.
15
Find a + b
Find a ­ b
b
a = 5i + 8j + 7k
b = ­2.5i + 0j ­7k
.
16
Magnitude of a vector
Magnitude is the length of a vector.
­ found using Pythagorean Thm
or
­ distance formula (pyth thm)
x
if a = xi + yj = y
x
a = xi + yj + zk = y
z
then a = √x2+y2
a = √x2 + y2 + z2
( )
( )
.
17
3
Find the length of the vector a = . Express your ­1
answer as a surd.
2
( )
Ans: √14 18
Unit Vectors
A unit vector is a vector with a length of 1.
( )
( )
ex. 1 = i + 0j
0
is a unit vector
0
ex. = 0i + j is a unit vector
1
Unit vectors are not always vertical or horizontal though...
19
Find the components of a unit vector in the direction of a = 5i ­ 2j.
20
a = a a
To find a unit vector in the direction of another vector:
­ find the length of the vector given
­ divide the components by that amount
This proportionally reduces the components to the length that would result in a unit vector.
21
Find a vector with length of 3 in the direction of a = i ­ j + 2k.
(similar process...find a unit vector in the same direction first, then multiply the components by 3)
22
23