Triangle law of vector addition

You’re a tourist in London and want to travel Westminster to Green Park.

How do you get there?

TFL UPDATE: Jubilee Line is Down due to engineering works.

Using the tube how do you reach Green Park now?

Let the District line from Westminster (W) to Victoria (V) be the vector WV = w .

Green Park (G)

Victoria(V)

w

v

Westminster (W)

Let the Victoria line from Victoria (V) to Green Park (G) be the vector VG = v.

Let the Jubilee line from Westminster (W) to Green Park (G) be the vector WG = g.

g

Westminster to Green park = WG = g

Westminster to Green park = WV + VGand

So WG = WV + VG Then w + v = g

= w + v

Triangle Law of Vector Additi on

When c = a + b the vector c is said to be the RESULTANT of the two vectors a and b.

By the Triangle Law of Vector Addition:

AB + BC = AC

a + b = c

A fellow tourist in London asks you how to get from Green Park to South Kensington.

How do you get there?

TFL UPDATE: Piccadilly Line is shut due to broken down train.

Using the tube how do you reach South Kensington now?

Let Green Park (G) to Victoria (V) be the vector GV = g .

Green Park (G)

Victoria(V)

v

g

Let Victoria (V) to South Ken (K) be the vector VK = v.

Let Green Park (G) to South Ken (K) be the vector GK = k.

Green Park to South Kensington = GK = k

Green Park to South Kensington = GV + VKand

So GK = GV + VK Then g + v = k

South Ken (K)

k

= g + v

WHICH TWO WAYS GET YOU GET FROM BANK TO LIVERPOOL STREET?

Let Bank (B) to Moorgate (M) be the vector BM = b

Liverpool Street (L)

Moorgate (M)

b

m

Let Moorgate (M) to Liverpool Street (L) be the vector MV = m

Let Bank (B) to Liverpool Street (L) be the vector BL = l

Bank to Liverpool Street = BL = l

Bank to Liverpool Street = BM + MLand

So BL = BM + ML Then b + m = l

Bank (B)

l

So

= b + m

AC = AB + BCAC = a + b

AD = AC + CDAD = a + b + c

i) AB = AO + OB AB = -a + b = b - a

ii) AP = ½ AB AP = ½ ( b – a)

ii) OP = ½ AB + OA OP = ½ ( b – a) + a

The Triangle Law of Vector Addition

Adding two vectors is equivalent to applying one vector followed by the other. For example,

Suppose a =5

3and b =

3

–2

Find a + b

We can represent this addition in the following diagram:

ab

a + b

a + b =8

1

Adding Vectors

When two or more vectors are added together the result is called the resultant vector.

In general, if a =a

band b =

c

d

We can add two column vectors by adding the horizontal components together and adding the vertical components together.

a + b =a + c

b + d

Adding Vectors

Subtracting Vectors

We can subtract two column vectors by subtracting the horizontal components and subtracting the vertical components. For example,

Find a – b

Suppose and b =–2

3a =

4

4

a – b =4

4–

–2

3=

4 – –2

4 – 3=

6

1

Subtracting Vectors

To show this subtraction in a diagram, we can think of a – b as a + (–b).

and b =–2

3a =

4

4

ab

a – b

a – b =6

1

–b a –b

Adding and Subtracting Vectors