February 2, 2010
MCV4U
Addition and Subtraction of Vectors
When you add two or more vectors, you are finding the resultant.
If you walk north 800 m, then west 600 m, the result is the same as if you walked 1000 m
[N37°W].
600 m
800 m
1000 m
37
The resultant vector is often represented by the symbol 𝑅⃗ .
Vector Addition
The addition of two vectors may be illustrated by using the triangle law of vector addition.
Consider the vectors 𝑎 and 𝑏⃗.
b
a
Place 𝑏⃗ so that its tail touches the tip of 𝑎.
February 2, 2010
b
a
Draw a new vector from the tail of 𝑎 to the tip of 𝑏⃗. This new vector is the resultant vector 𝑎 + 𝑏⃗.
b
a
R
This is called the triangle method, or tail-to-tip method, of adding vectors.
If the vectors are arranged tail-to-tail, another method can be used.
Consider the same two vectors, 𝑎 and 𝑏⃗, and arrange them tail-to-tail.
a
b
Complete a parallelogram that has 𝑎 and 𝑏⃗ as two of the sides.
a
b
The diagonal of this parallelogram is the resultant 𝑎 + 𝑏⃗.
February 2, 2010
a
R
b
This is called the parallelogram method, or the tail-to-tail method, of vector addition.
Vector Subtraction
Once you can add vectors, subtraction is simple. To find the resultant 𝑎 − 𝑏⃗, simply add 𝑎 to the
opposite of 𝑏⃗, or -𝑏⃗.
Example:
For the vectors 𝑢
⃗ and 𝑣 below, find 𝑢
⃗ + 𝑣 and 𝑢
⃗ − 𝑣.
v
u
a) find 𝑢
⃗ +𝑣
Triangle method:
u+v
u
v
February 2, 2010
Parallelogram method:
v
u+v
u
b) Find 𝑢
⃗ −𝑣
Head-to-tail method:
u
u-v
-v
Parallelogram method:
-v
u-v
u
Tail-to-tail method:
v
u
u-v