VECTOR ADDITION
(Geometric Vectors)
A.
THE RESULTANT VECTOR
The resultant of two vectors is formed by joining the initial point of the first vector
to the terminal point of the second vector.
B
𝐴𝐵 + 𝐵𝐶 = 𝐴𝐶
A
B.
resultant
C
VECTOR ADDITION
1.
Triangle Law of Vector Addition
𝑢
𝑢+𝑣
𝑣
𝑣
𝑣+𝑢
𝑢
NOTE:
2.
𝑢+𝑣 =𝑣+𝑢
Parallelogram Law of Vector Addition
𝑢
𝑣
𝑢
𝑢+𝑣
𝑣
𝑢
𝑢 − 𝑣 = 𝑢 + −𝑣
TRIANGLE INEQUALITY
𝑢+𝑣 ≤ 𝑢 + 𝑣
−𝑣
𝑢
NOTE:
C.
𝑢−𝑣
𝑣
In what case does equality hold?
D.
THE ZERO VECTOR
When two opposite vectors are added, the resultant is the zero vector, 0.
The zero vector has a magnitude of 0 (ie) 0 = 0, and no defined direction.
B
A
E.
B
𝐴𝐵 + 𝐵𝐴 = 0
A
THE ANGLE BETWEEN TWO VECTORS
The angle between two vectors, , is formed when the vectors are placed tail to tail.
𝑢
NOTE:
0    180o

𝑣
F.
EXAMPLES
1.
Write each vector as a sum/difference:
𝑏
𝑎
𝑎=
𝑏=
𝑐=
𝑐
2.
Write as a single equivalent vector:
a)
𝐴𝐵 + 𝐵𝐶 + 𝐶𝐷 =
b)
𝐴𝐵 − 𝐺𝐶 + 𝐵𝐶 − 𝐴𝐺 =
3.
4.
In the given rectangular box, 𝑂𝐴 = 𝑎, 𝑂𝐶 = 𝑏, and 𝑂𝐷 = 𝑐. Express each of
the following in terms of 𝑎, 𝑏, and 𝑐 :
𝐵𝐶 =
b)
𝐺𝐸 =
c)
𝑂𝐹 =
Given vectors, 𝑎, 𝑏, and 𝑐, shown below, draw a vector equivalent to:
a)
5.
a)
𝑎+𝑏+𝑐
b)
𝑎−𝑏+𝑐
If 𝑢 = 2, 𝑣 = 3, and the angle between 𝑢 and 𝑣 is 35o, calculate 𝑢 + 𝑣 .
Homework: p.290–292 #1–4, 5(no sketch), 6, 7, 9, 12–15