Vector Addition

And Distance vs. Displacement

Distance and Displacement

• In Physics there is a difference between distance and displacement.

• Distance: The length of the route travelled between two points.

• Displacement: The shortest path between two points.

Distance vs. Displacement• Look at the following map:

Distance vs. Displacement• The Distance is how many km the boat travels to get to the treasure…

Distance vs. Displacement• The Displacement is how many km it is from the ship to the treasure in a

straight line:

Distance vs. Displacement• Distance is often larger than Displacement. Sometimes they may be

equal, though.

Distance vs. Displacement

• Distance > Displacement.

Distance vs. Displacement•Distance = Displacement

Displacement & Vectors

• Vectors can be connected together to create a “map” similar to the treasure map:

Displacement & Vectors

• Connecting vectors like this is called “Vector Addition”:

Displacement & Vectors• When vectors are added together the

“answer” is the displacement. Adding vectors together can also be called “resolving vectors”

Adding Vectors

• Resolve the following vectors: 3cm due N and 3cm due East.

N Start the first vector from the origin.

W E

S

3cm

Adding Vectors

• Resolve the following vectors: 3cm due N and 3cm due East.

N Start the second vector from the end

of the first

W E

S

3cm

3cm

Adding Vectors

• Resolve the following vectors: 3cm due N and 3cm due East.

N Draw the resultant and

calculate the length/angle.

W E

S

3cm

3cm

Adding Vectors

• Resolve the following vectors: 3cm due N and 3cm due East.

N The length can be foundusing the PythagoreanTheorem: a2 + b2 = c2

W E

S

3cm

3cm

Adding Vectors

• Resolve the following vectors: 3cm due N and 3cm due East.

N a2 + b2 = c2

(3cm)2 + (3cm)2 = c2

W E

S

3cm (b)

3cm

(a)

c

Adding Vectors

• Resolve the following vectors: 3cm due N and 3cm due East.

N a2 + b2 = c2

(3cm)2 + (3cm)2 = c2

18 cm2 = c2

4.24 cm = cW E

S

3cm (b)

3cm

(a)

c

Adding Vectors

• Resolve the following vectors: 3cm due N and 3cm due East.

N The angle can be found using what we know

abouttriangles and trig.

W E

S

3cm (b)

3cm

(a)

c = 4.24 cm

Adding Vectors

• Resolve the following vectors: 3cm due N and 3cm due East.

N First, lets find this angle:

.

W E

S

3cm

3cm

4.24 cm

Adding Vectors

• Resolve the following vectors: 3cm due N and 3cm due East.

N To get the angle we useone of the trig identities:sinθ

W E

S

3cm

3cm

4.24 cmθ

Adding Vectors

• Resolve the following vectors: 3cm due N and 3cm due East.

N

W E

S

3cm

3cm

4.24 cmθ

45

7075.0sin24.4

3sin

sin

cm

cm

hypotenuse

opposite

Adding Vectors

• Resolve the following vectors: 3cm due N and 3cm due East.

N But remember, the vectorangle is measured from

theEast-West Axis:

W E

S

3cm

3cm

4.24 cmΘ=45˚

Adding Vectors

• Resolve the following vectors: 3cm due N and 3cm due East.

N Since the angle betweenNorth and East is 90˚, wecan say:90˚-45˚=45˚

W E

S

3cm

3cm

4.24 cmΘ=45˚

Adding Vectors

• Resolve the following vectors: 3cm due N and 3cm due East.

N Now we have the lengthand the angle so:Answer: 4.24cm, 45˚NE

W E

S

3cm

3cm

4.24 cm45˚Θ=45˚

Adding Vectors

• Resolve the following vectors: 3cm due N and 3cm due East.

N Review: If you walked 3 cm Nand 3 cm E, what is the

distancetraveled?Displacement?

W E

S

3cm (b)

3cm

(a)

c = 4.24 cm

Adding Vectors

• Resolve the following vectors: 3cm due N and 3cm due East.

N Review: If you walked 3 cm Nand 3 cm E, what is the

distancetraveled? 3cm + 3cm = 6 cmDisplacement? 4.24 cm

W E

S

3cm (b)

3cm

(a)

c = 4.24 cm